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| | <div style="-moz-column-count:2; column-count:2;"> |
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| | ; Notation : [[24.1|24.1 Special Notation]]<br> |
| ! DLMF !! Formula !! Maple !! Mathematica !! Symbolic<br>Maple !! Symbolic<br>Mathematica !! Numeric<br>Maple !! Numeric<br>Mathematica
| | ; Properties : [[24.2|24.2 Definitions and Generating Functions]]<br>[[24.3|24.3 Graphs]]<br>[[24.4|24.4 Basic Properties]]<br>[[24.5|24.5 Recurrence Relations]]<br>[[24.6|24.6 Explicit Formulas]]<br>[[24.7|24.7 Integral Representations]]<br>[[24.8|24.8 Series Expansions]]<br>[[24.9|24.9 Inequalities]]<br>[[24.10|24.10 Arithmetic Properties]]<br>[[24.11|24.11 Asymptotic Approximations]]<br>[[24.12|24.12 Zeros]]<br>[[24.13|24.13 Integrals]]<br>[[24.14|24.14 Sums]]<br>[[24.15|24.15 Related Sequences of Numbers]]<br>[[24.16|24.16 Generalizations]]<br> |
| |-
| | ; Applications : [[24.17|24.17 Mathematical Applications]]<br>[[24.18|24.18 Physical Applications]]<br> |
| | [https://dlmf.nist.gov/24.1#Ex1 24.1#Ex1] || [[Item:Q7394|<math>B_{1} = \tfrac{1}{6}</math>]] || <code>B[1] = (1)/(6)</code> || <code>Subscript[B, 1] == Divide[1,6]</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
| | ; Computation : [[24.19|24.19 Methods of Computation]]<br>[[24.20|24.20 Tables]]<br>[[24.21|24.21 Software]]<br> |
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| | </div> |
| | [https://dlmf.nist.gov/24.1#Ex2 24.1#Ex2] || [[Item:Q7395|<math>B_{2} = \tfrac{1}{30}</math>]] || <code>B[2] = (1)/(30)</code> || <code>Subscript[B, 2] == Divide[1,30]</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| | [https://dlmf.nist.gov/24.1#Ex3 24.1#Ex3] || [[Item:Q7396|<math>B_{3} = \tfrac{1}{42}</math>]] || <code>B[3] = (1)/(42)</code> || <code>Subscript[B, 3] == Divide[1,42]</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| |-
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| | [https://dlmf.nist.gov/24.2.E1 24.2.E1] || [[Item:Q7398|<math>\frac{t}{e^{t}-1} = \sum_{n=0}^{\infty}\BernoullinumberB{n}\frac{t^{n}}{n!}</math>]] || <code>(t)/(exp(t)- 1) = sum(bernoulli(n)*((t)^(n))/(factorial(n)), n = 0..infinity)</code> || <code>Divide[t,Exp[t]- 1] == Sum[BernoulliB[n]*Divide[(t)^(n),(n)!], {n, 0, Infinity}, GenerateConditions->None]</code> || Failure || Successful || Successful [Tested: 6] || Successful [Tested: 6]
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| |
| | [https://dlmf.nist.gov/24.2#Ex1 24.2#Ex1] || [[Item:Q7399|<math>\BernoullinumberB{2n+1} = 0</math>]] || <code>bernoulli(2*n + 1) = 0</code> || <code>BernoulliB[2*n + 1] == 0</code> || Failure || Failure || Successful [Tested: 3] || Successful [Tested: 3]
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| | [https://dlmf.nist.gov/24.2#Ex2 24.2#Ex2] || [[Item:Q7400|<math>(-1)^{n+1}\BernoullinumberB{2n} > 0</math>]] || <code>(- 1)^(n + 1)* bernoulli(2*n) > 0</code> || <code>(- 1)^(n + 1)* BernoulliB[2*n] > 0</code> || Failure || Failure || Successful [Tested: 1] || Successful [Tested: 3]
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| |
| | [https://dlmf.nist.gov/24.2.E3 24.2.E3] || [[Item:Q7401|<math>\frac{te^{xt}}{e^{t}-1} = \sum_{n=0}^{\infty}\BernoullipolyB{n}@{x}\frac{t^{n}}{n!}</math>]] || <code>(t*exp(x*t))/(exp(t)- 1) = sum(bernoulli(n, x)*((t)^(n))/(factorial(n)), n = 0..infinity)</code> || <code>Divide[t*Exp[x*t],Exp[t]- 1] == Sum[BernoulliB[n, x]*Divide[(t)^(n),(n)!], {n, 0, Infinity}, GenerateConditions->None]</code> || Failure || Successful || Successful [Tested: 18] || Successful [Tested: 18]
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| |
| | [https://dlmf.nist.gov/24.2.E4 24.2.E4] || [[Item:Q7402|<math>\BernoullinumberB{n} = \BernoullipolyB{n}@{0}</math>]] || <code>bernoulli(n) = bernoulli(n, 0)</code> || <code>BernoulliB[n] == BernoulliB[n, 0]</code> || Successful || Successful || - || Successful [Tested: 3]
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| |
| | [https://dlmf.nist.gov/24.2.E5 24.2.E5] || [[Item:Q7403|<math>\BernoullipolyB{n}@{x} = \sum_{k=0}^{n}{n\choose k}\BernoullinumberB{k}x^{n-k}</math>]] || <code>bernoulli(n, x) = sum(binomial(n,k)*bernoulli(k)*(x)^(n - k), k = 0..n)</code> || <code>BernoulliB[n, x] == Sum[Binomial[n,k]*BernoulliB[k]*(x)^(n - k), {k, 0, n}, GenerateConditions->None]</code> || Failure || Successful || Successful [Tested: 9] || Successful [Tested: 9]
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| |
| | [https://dlmf.nist.gov/24.2.E6 24.2.E6] || [[Item:Q7404|<math>\frac{2e^{t}}{e^{2t}+1} = \sum_{n=0}^{\infty}\EulernumberE{n}\frac{t^{n}}{n!}</math>]] || <code>Error</code> || <code>Divide[2*Exp[t],Exp[2*t]+ 1] == Sum[EulerE[n]*Divide[(t)^(n),(n)!], {n, 0, Infinity}, GenerateConditions->None]</code> || Missing Macro Error || Successful || - || Successful [Tested: 4]
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| | [https://dlmf.nist.gov/24.2#Ex3 24.2#Ex3] || [[Item:Q7405|<math>\EulernumberE{2n+1} = 0</math>]] || <code>Error</code> || <code>EulerE[2*n + 1] == 0</code> || Missing Macro Error || Failure || - || Successful [Tested: 3]
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| | [https://dlmf.nist.gov/24.2#Ex4 24.2#Ex4] || [[Item:Q7406|<math>(-1)^{n}\EulernumberE{2n} > 0</math>]] || <code>Error</code> || <code>(- 1)^(n)* EulerE[2*n] > 0</code> || Missing Macro Error || Failure || - || Successful [Tested: 3]
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| | [https://dlmf.nist.gov/24.2.E8 24.2.E8] || [[Item:Q7407|<math>\frac{2e^{xt}}{e^{t}+1} = \sum_{n=0}^{\infty}\EulerpolyE{n}@{x}\frac{t^{n}}{n!}</math>]] || <code>(2*exp(x*t))/(exp(t)+ 1) = sum(euler(n, x)*((t)^(n))/(factorial(n)), n = 0..infinity)</code> || <code>Divide[2*Exp[x*t],Exp[t]+ 1] == Sum[EulerE[n, x]*Divide[(t)^(n),(n)!], {n, 0, Infinity}, GenerateConditions->None]</code> || Failure || Successful || Error || Successful [Tested: 18]
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| | [https://dlmf.nist.gov/24.2.E9 24.2.E9] || [[Item:Q7408|<math>\EulernumberE{n} = 2^{n}\EulerpolyE{n}@{\tfrac{1}{2}}</math>]] || <code>Error</code> || <code>EulerE[n] == (2)^(n)* EulerE[n, Divide[1,2]]</code> || Missing Macro Error || Successful || - || Successful [Tested: 3]
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| | [https://dlmf.nist.gov/24.2.E10 24.2.E10] || [[Item:Q7409|<math>\EulerpolyE{n}@{x} = \sum_{k=0}^{n}{n\choose k}\frac{\EulernumberE{k}}{2^{k}}(x-\tfrac{1}{2})^{n-k}</math>]] || <code>Error</code> || <code>EulerE[n, x] == Sum[Binomial[n,k]*Divide[EulerE[k],(2)^(k)]*(x -Divide[1,2])^(n - k), {k, 0, n}, GenerateConditions->None]</code> || Missing Macro Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [3 / 9]<div class="mw-collapsible-content"><code>{Indeterminate <- {Rule[n, 1], Rule[x, 0.5]}</code><br><code>Indeterminate <- {Rule[n, 2], Rule[x, 0.5]}</code><br></div></div>
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| | [https://dlmf.nist.gov/24.4.E1 24.4.E1] || [[Item:Q7414|<math>\BernoullipolyB{n}@{x+1}-\BernoullipolyB{n}@{x} = nx^{n-1}</math>]] || <code>bernoulli(n, x + 1)- bernoulli(n, x) = n*(x)^(n - 1)</code> || <code>BernoulliB[n, x + 1]- BernoulliB[n, x] == n*(x)^(n - 1)</code> || Successful || Successful || - || Successful [Tested: 9]
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| |
| | [https://dlmf.nist.gov/24.4.E2 24.4.E2] || [[Item:Q7415|<math>\EulerpolyE{n}@{x+1}+\EulerpolyE{n}@{x} = 2x^{n}</math>]] || <code>euler(n, x + 1)+ euler(n, x) = 2*(x)^(n)</code> || <code>EulerE[n, x + 1]+ EulerE[n, x] == 2*(x)^(n)</code> || Successful || Successful || - || Successful [Tested: 9]
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| | [https://dlmf.nist.gov/24.4.E3 24.4.E3] || [[Item:Q7416|<math>\BernoullipolyB{n}@{1-x} = (-1)^{n}\BernoullipolyB{n}@{x}</math>]] || <code>bernoulli(n, 1 - x) = (- 1)^(n)* bernoulli(n, x)</code> || <code>BernoulliB[n, 1 - x] == (- 1)^(n)* BernoulliB[n, x]</code> || Successful || Failure || - || Successful [Tested: 9]
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| |
| | [https://dlmf.nist.gov/24.4.E4 24.4.E4] || [[Item:Q7417|<math>\EulerpolyE{n}@{1-x} = (-1)^{n}\EulerpolyE{n}@{x}</math>]] || <code>euler(n, 1 - x) = (- 1)^(n)* euler(n, x)</code> || <code>EulerE[n, 1 - x] == (- 1)^(n)* EulerE[n, x]</code> || Successful || Failure || - || Successful [Tested: 9]
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| | [https://dlmf.nist.gov/24.4.E5 24.4.E5] || [[Item:Q7418|<math>(-1)^{n}\BernoullipolyB{n}@{-x} = \BernoullipolyB{n}@{x}+nx^{n-1}</math>]] || <code>(- 1)^(n)* bernoulli(n, - x) = bernoulli(n, x)+ n*(x)^(n - 1)</code> || <code>(- 1)^(n)* BernoulliB[n, - x] == BernoulliB[n, x]+ n*(x)^(n - 1)</code> || Successful || Failure || - || Successful [Tested: 9]
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| | [https://dlmf.nist.gov/24.4.E6 24.4.E6] || [[Item:Q7419|<math>(-1)^{n+1}\EulerpolyE{n}@{-x} = \EulerpolyE{n}@{x}-2x^{n}</math>]] || <code>(- 1)^(n + 1)* euler(n, - x) = euler(n, x)- 2*(x)^(n)</code> || <code>(- 1)^(n + 1)* EulerE[n, - x] == EulerE[n, x]- 2*(x)^(n)</code> || Successful || Failure || - || Successful [Tested: 9]
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| | [https://dlmf.nist.gov/24.4.E7 24.4.E7] || [[Item:Q7420|<math>\sum_{k=1}^{m}k^{n} = \frac{\BernoullipolyB{n+1}@{m+1}-\BernoullinumberB{n+1}}{n+1}</math>]] || <code>sum((k)^(n), k = 1..m) = (bernoulli(n + 1, m + 1)- bernoulli(n + 1))/(n + 1)</code> || <code>Sum[(k)^(n), {k, 1, m}, GenerateConditions->None] == Divide[BernoulliB[n + 1, m + 1]- BernoulliB[n + 1],n + 1]</code> || Failure || Failure || Successful [Tested: 9] || Successful [Tested: 9]
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| | [https://dlmf.nist.gov/24.4.E8 24.4.E8] || [[Item:Q7421|<math>\sum_{k=1}^{m}(-1)^{m-k}k^{n} = \frac{\EulerpolyE{n}@{m+1}+(-1)^{m}\EulerpolyE{n}@{0}}{2}</math>]] || <code>sum((- 1)^(m - k)* (k)^(n), k = 1..m) = (euler(n, m + 1)+(- 1)^(m)* euler(n, 0))/(2)</code> || <code>Sum[(- 1)^(m - k)* (k)^(n), {k, 1, m}, GenerateConditions->None] == Divide[EulerE[n, m + 1]+(- 1)^(m)* EulerE[n, 0],2]</code> || Failure || Failure || Successful [Tested: 9] || Successful [Tested: 9]
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| | [https://dlmf.nist.gov/24.4.E9 24.4.E9] || [[Item:Q7422|<math>\sum_{k=0}^{m-1}(a+dk)^{n} = {\frac{d^{n}}{n+1}\left(\BernoullipolyB{n+1}@{m+\frac{a}{d}}-\BernoullipolyB{n+1}@{\frac{a}{d}}\right)}</math>]] || <code>sum((a + d*k)^(n), k = 0..m - 1) = ((d)^(n))/(n + 1)*(bernoulli(n + 1, m +(a)/(d))- bernoulli(n + 1, (a)/(d)))</code> || <code>Sum[(a + d*k)^(n), {k, 0, m - 1}, GenerateConditions->None] == Divide[(d)^(n),n + 1]*(BernoulliB[n + 1, m +Divide[a,d]]- BernoulliB[n + 1, Divide[a,d]])</code> || Failure || Failure || Successful [Tested: 300] || Successful [Tested: 300]
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| | [https://dlmf.nist.gov/24.4.E10 24.4.E10] || [[Item:Q7423|<math>\sum_{k=0}^{m-1}(-1)^{k}(a+dk)^{n} = {\frac{d^{n}}{2}\left((-1)^{m-1}\EulerpolyE{n}@{m+\frac{a}{d}}+\EulerpolyE{n}@{\frac{a}{d}}\right)}</math>]] || <code>sum((- 1)^(k)*(a + d*k)^(n), k = 0..m - 1) = ((d)^(n))/(2)*((- 1)^(m - 1)* euler(n, m +(a)/(d))+ euler(n, (a)/(d)))</code> || <code>Sum[(- 1)^(k)*(a + d*k)^(n), {k, 0, m - 1}, GenerateConditions->None] == Divide[(d)^(n),2]*((- 1)^(m - 1)* EulerE[n, m +Divide[a,d]]+ EulerE[n, Divide[a,d]])</code> || Failure || Failure || Successful [Tested: 300] || Successful [Tested: 300]
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| | [https://dlmf.nist.gov/24.4.E12 24.4.E12] || [[Item:Q7425|<math>\BernoullipolyB{n}@{x+h} = \sum_{k=0}^{n}{n\choose k}\BernoullipolyB{k}@{x}h^{n-k}</math>]] || <code>bernoulli(n, x + h) = sum(binomial(n,k)*bernoulli(k, x)*(h)^(n - k), k = 0..n)</code> || <code>BernoulliB[n, x + h] == Sum[Binomial[n,k]*BernoulliB[k, x]*(h)^(n - k), {k, 0, n}, GenerateConditions->None]</code> || Failure || Failure || Successful [Tested: 90] || Successful [Tested: 90]
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| | [https://dlmf.nist.gov/24.4.E13 24.4.E13] || [[Item:Q7426|<math>\EulerpolyE{n}@{x+h} = \sum_{k=0}^{n}{n\choose k}\EulerpolyE{k}@{x}h^{n-k}</math>]] || <code>euler(n, x + h) = sum(binomial(n,k)*euler(k, x)*(h)^(n - k), k = 0..n)</code> || <code>EulerE[n, x + h] == Sum[Binomial[n,k]*EulerE[k, x]*(h)^(n - k), {k, 0, n}, GenerateConditions->None]</code> || Failure || Failure || Successful [Tested: 90] || Successful [Tested: 90]
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| | [https://dlmf.nist.gov/24.4.E14 24.4.E14] || [[Item:Q7427|<math>\EulerpolyE{n-1}@{x} = \frac{2}{n}\sum_{k=0}^{n}{n\choose k}(1-2^{k})\BernoullinumberB{k}x^{n-k}</math>]] || <code>euler(n - 1, x) = (2)/(n)*sum(binomial(n,k)*(1 - (2)^(k))* bernoulli(k)*(x)^(n - k), k = 0..n)</code> || <code>EulerE[n - 1, x] == Divide[2,n]*Sum[Binomial[n,k]*(1 - (2)^(k))* BernoulliB[k]*(x)^(n - k), {k, 0, n}, GenerateConditions->None]</code> || Failure || Failure || Successful [Tested: 9] || Successful [Tested: 9]
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| | [https://dlmf.nist.gov/24.4.E15 24.4.E15] || [[Item:Q7428|<math>\BernoullinumberB{2n} = \frac{2n}{2^{2n}(2^{2n}-1)}\sum_{k=0}^{n-1}{2n-1\choose 2k}\EulernumberE{2k}</math>]] || <code>Error</code> || <code>BernoulliB[2*n] == Divide[2*n,(2)^(2*n)*((2)^(2*n)- 1)]*Sum[Binomial[2*n - 1,2*k]*EulerE[2*k], {k, 0, n - 1}, GenerateConditions->None]</code> || Missing Macro Error || Failure || - || Successful [Tested: 3]
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| | [https://dlmf.nist.gov/24.4.E16 24.4.E16] || [[Item:Q7429|<math>\EulernumberE{2n} = \frac{1}{2n+1}-\sum_{k=1}^{n}{2n\choose 2k-1}\frac{2^{2k}(2^{2k-1}-1)\BernoullinumberB{2k}}{k}</math>]] || <code>Error</code> || <code>EulerE[2*n] == Divide[1,2*n + 1]- Sum[Binomial[2*n,2*k - 1]*Divide[(2)^(2*k)*((2)^(2*k - 1)- 1)* BernoulliB[2*k],k], {k, 1, n}, GenerateConditions->None]</code> || Missing Macro Error || Aborted || - || Skipped - Because timed out
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| | [https://dlmf.nist.gov/24.4.E17 24.4.E17] || [[Item:Q7430|<math>\EulernumberE{2n} = 1-\sum_{k=1}^{n}{2n\choose 2k-1}\frac{2^{2k}(2^{2k}-1)\BernoullinumberB{2k}}{2k}</math>]] || <code>Error</code> || <code>EulerE[2*n] == 1 - Sum[Binomial[2*n,2*k - 1]*Divide[(2)^(2*k)*((2)^(2*k)- 1)* BernoulliB[2*k],2*k], {k, 1, n}, GenerateConditions->None]</code> || Missing Macro Error || Aborted || - || Skipped - Because timed out
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| | [https://dlmf.nist.gov/24.4.E18 24.4.E18] || [[Item:Q7431|<math>\BernoullipolyB{n}@{mx} = m^{n-1}\sum_{k=0}^{m-1}\BernoullipolyB{n}@{x+\frac{k}{m}}</math>]] || <code>bernoulli(n, m*x) = (m)^(n - 1)* sum(bernoulli(n, x +(k)/(m)), k = 0..m - 1)</code> || <code>BernoulliB[n, m*x] == (m)^(n - 1)* Sum[BernoulliB[n, x +Divide[k,m]], {k, 0, m - 1}, GenerateConditions->None]</code> || Failure || Successful || Successful [Tested: 27] || Successful [Tested: 27]
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| | [https://dlmf.nist.gov/24.4.E19 24.4.E19] || [[Item:Q7432|<math>\EulerpolyE{n}@{mx} = -\frac{2m^{n}}{n+1}\sum_{k=0}^{m-1}(-1)^{k}\BernoullipolyB{n+1}@{x+\frac{k}{m}}</math>]] || <code>euler(n, m*x) = -(2*(m)^(n))/(n + 1)*sum((- 1)^(k)* bernoulli(n + 1, x +(k)/(m)), k = 0..m - 1)</code> || <code>EulerE[n, m*x] == -Divide[2*(m)^(n),n + 1]*Sum[(- 1)^(k)* BernoulliB[n + 1, x +Divide[k,m]], {k, 0, m - 1}, GenerateConditions->None]</code> || Failure || Failure || Successful [Tested: 9] || <div class="toccolours mw-collapsible mw-collapsed">Failed [18 / 27]<div class="mw-collapsible-content"><code>{1.9166666666666667 <- {Rule[m, 1], Rule[n, 1], Rule[x, 1.5]}</code><br><code>1.25 <- {Rule[m, 1], Rule[n, 2], Rule[x, 1.5]}</code><br></div></div>
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| | [https://dlmf.nist.gov/24.4.E20 24.4.E20] || [[Item:Q7433|<math>\EulerpolyE{n}@{mx} = m^{n}\sum_{k=0}^{m-1}(-1)^{k}\EulerpolyE{n}@{x+\frac{k}{m}}</math>]] || <code>euler(n, m*x) = (m)^(n)* sum((- 1)^(k)* euler(n, x +(k)/(m)), k = 0..m - 1)</code> || <code>EulerE[n, m*x] == (m)^(n)* Sum[(- 1)^(k)* EulerE[n, x +Divide[k,m]], {k, 0, m - 1}, GenerateConditions->None]</code> || Failure || Failure || Successful [Tested: 9] || <div class="toccolours mw-collapsible mw-collapsed">Failed [9 / 27]<div class="mw-collapsible-content"><code>{3.5 <- {Rule[m, 2], Rule[n, 1], Rule[x, 1.5]}</code><br><code>11.0 <- {Rule[m, 2], Rule[n, 2], Rule[x, 1.5]}</code><br></div></div>
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| | [https://dlmf.nist.gov/24.4.E21 24.4.E21] || [[Item:Q7434|<math>\BernoullipolyB{n}@{x} = 2^{n-1}\left(\BernoullipolyB{n}@{\tfrac{1}{2}x}+\BernoullipolyB{n}@{\tfrac{1}{2}x+\tfrac{1}{2}}\right)</math>]] || <code>bernoulli(n, x) = (2)^(n - 1)*(bernoulli(n, (1)/(2)*x)+ bernoulli(n, (1)/(2)*x +(1)/(2)))</code> || <code>BernoulliB[n, x] == (2)^(n - 1)*(BernoulliB[n, Divide[1,2]*x]+ BernoulliB[n, Divide[1,2]*x +Divide[1,2]])</code> || Failure || Failure || Successful [Tested: 9] || Successful [Tested: 9]
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| | [https://dlmf.nist.gov/24.4.E22 24.4.E22] || [[Item:Q7435|<math>\EulerpolyE{n-1}@{x} = \frac{2}{n}\left(\BernoullipolyB{n}@{x}-2^{n}\BernoullipolyB{n}@{\tfrac{1}{2}x}\right)</math>]] || <code>euler(n - 1, x) = (2)/(n)*(bernoulli(n, x)- (2)^(n)* bernoulli(n, (1)/(2)*x))</code> || <code>EulerE[n - 1, x] == Divide[2,n]*(BernoulliB[n, x]- (2)^(n)* BernoulliB[n, Divide[1,2]*x])</code> || Failure || Failure || Successful [Tested: 9] || Successful [Tested: 9]
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| | [https://dlmf.nist.gov/24.4.E23 24.4.E23] || [[Item:Q7436|<math>\EulerpolyE{n-1}@{x} = \frac{2^{n}}{n}\left(\BernoullipolyB{n}@{\tfrac{1}{2}x+\tfrac{1}{2}}-\BernoullipolyB{n}@{\tfrac{1}{2}x}\right)</math>]] || <code>euler(n - 1, x) = ((2)^(n))/(n)*(bernoulli(n, (1)/(2)*x +(1)/(2))- bernoulli(n, (1)/(2)*x))</code> || <code>EulerE[n - 1, x] == Divide[(2)^(n),n]*(BernoulliB[n, Divide[1,2]*x +Divide[1,2]]- BernoulliB[n, Divide[1,2]*x])</code> || Failure || Failure || Successful [Tested: 9] || Successful [Tested: 9]
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| | [https://dlmf.nist.gov/24.4.E24 24.4.E24] || [[Item:Q7437|<math>\BernoullipolyB{n}@{mx} = m^{n}\BernoullipolyB{n}@{x}+n\sum_{k=1}^{n}\sum_{j=0}^{k-1}(-1)^{j}{n\choose k}\*\left(\sum_{r=1}^{m-1}\frac{e^{2\pi i(k-j)r/m}}{(1-e^{2\pi ir/m})^{n}}\right)(j+mx)^{n-1}</math>]] || <code>bernoulli(n, m*x) = (m)^(n)* bernoulli(n, x)+ n*sum(sum((- 1)^(j)*binomial(n,k)*(sum((exp(2*Pi*I*(k - j)* r/ m))/((1 - exp(2*Pi*I*r/ m))^(n)), r = 1..m - 1))*(j + m*x)^(n - 1), j = 0..k - 1), k = 1..n)</code> || <code>BernoulliB[n, m*x] == (m)^(n)* BernoulliB[n, x]+ n*Sum[Sum[(- 1)^(j)*Binomial[n,k]*(Sum[Divide[Exp[2*Pi*I*(k - j)* r/ m],(1 - Exp[2*Pi*I*r/ m])^(n)], {r, 1, m - 1}, GenerateConditions->None])*(j + m*x)^(n - 1), {j, 0, k - 1}, GenerateConditions->None], {k, 1, n}, GenerateConditions->None]</code> || Aborted || Failure || Skipped - Because timed out || <div class="toccolours mw-collapsible mw-collapsed">Failed [17 / 27]<div class="mw-collapsible-content"><code>{1.0 <- {Rule[m, 2], Rule[n, 1], Rule[x, 1.5]}</code><br><code>1.9999999999999991 <- {Rule[m, 2], Rule[n, 2], Rule[x, 1.5]}</code><br></div></div>
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| | [https://dlmf.nist.gov/24.4.E25 24.4.E25] || [[Item:Q7438|<math>\BernoullipolyB{n}@{0} = (-1)^{n}\BernoullipolyB{n}@{1}</math>]] || <code>bernoulli(n, 0) = (- 1)^(n)* bernoulli(n, 1)</code> || <code>BernoulliB[n, 0] == (- 1)^(n)* BernoulliB[n, 1]</code> || Successful || Failure || - || Successful [Tested: 3]
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| |-
| |
| | [https://dlmf.nist.gov/24.4.E25 24.4.E25] || [[Item:Q7438|<math>(-1)^{n}\BernoullipolyB{n}@{1} = \BernoullinumberB{n}</math>]] || <code>(- 1)^(n)* bernoulli(n, 1) = bernoulli(n)</code> || <code>(- 1)^(n)* BernoulliB[n, 1] == BernoulliB[n]</code> || Successful || Failure || - || Successful [Tested: 3]
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| |-
| |
| | [https://dlmf.nist.gov/24.4.E26 24.4.E26] || [[Item:Q7439|<math>\EulerpolyE{n}@{0} = -\EulerpolyE{n}@{1}</math>]] || <code>euler(n, 0) = - euler(n, 1)</code> || <code>EulerE[n, 0] == - EulerE[n, 1]</code> || Successful || Successful || Skip - symbolical successful subtest || Successful [Tested: 3]
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| |-
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| | [https://dlmf.nist.gov/24.4.E26 24.4.E26] || [[Item:Q7439|<math>-\EulerpolyE{n}@{1} = -\frac{2}{n+1}(2^{n+1}-1)\BernoullinumberB{n+1}</math>]] || <code>- euler(n, 1) = -(2)/(n + 1)*((2)^(n + 1)- 1)* bernoulli(n + 1)</code> || <code>- EulerE[n, 1] == -Divide[2,n + 1]*((2)^(n + 1)- 1)* BernoulliB[n + 1]</code> || Failure || Failure || Successful [Tested: 3] || Successful [Tested: 3]
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| | [https://dlmf.nist.gov/24.4.E27 24.4.E27] || [[Item:Q7440|<math>\BernoullipolyB{n}@{\tfrac{1}{2}} = -(1-2^{1-n})\BernoullinumberB{n}</math>]] || <code>bernoulli(n, (1)/(2)) = -(1 - (2)^(1 - n))* bernoulli(n)</code> || <code>BernoulliB[n, Divide[1,2]] == -(1 - (2)^(1 - n))* BernoulliB[n]</code> || Successful || Successful || - || Successful [Tested: 3]
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| | [https://dlmf.nist.gov/24.4.E28 24.4.E28] || [[Item:Q7441|<math>\EulerpolyE{n}@{\tfrac{1}{2}} = 2^{-n}\EulernumberE{n}</math>]] || <code>Error</code> || <code>EulerE[n, Divide[1,2]] == (2)^(- n)* EulerE[n]</code> || Missing Macro Error || Successful || - || Successful [Tested: 3]
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| | [https://dlmf.nist.gov/24.4.E29 24.4.E29] || [[Item:Q7442|<math>\BernoullipolyB{2n}@{\tfrac{1}{3}} = \BernoullipolyB{2n}@{\tfrac{2}{3}}</math>]] || <code>bernoulli(2*n, (1)/(3)) = bernoulli(2*n, (2)/(3))</code> || <code>BernoulliB[2*n, Divide[1,3]] == BernoulliB[2*n, Divide[2,3]]</code> || Failure || Failure || Successful [Tested: 3] || Successful [Tested: 3]
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| | [https://dlmf.nist.gov/24.4.E29 24.4.E29] || [[Item:Q7442|<math>\BernoullipolyB{2n}@{\tfrac{2}{3}} = -\tfrac{1}{2}(1-3^{1-2n})\BernoullinumberB{2n}</math>]] || <code>bernoulli(2*n, (2)/(3)) = -(1)/(2)*(1 - (3)^(1 - 2*n))* bernoulli(2*n)</code> || <code>BernoulliB[2*n, Divide[2,3]] == -Divide[1,2]*(1 - (3)^(1 - 2*n))* BernoulliB[2*n]</code> || Failure || Failure || Successful [Tested: 3] || Successful [Tested: 3]
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| | [https://dlmf.nist.gov/24.4.E30 24.4.E30] || [[Item:Q7443|<math>\EulerpolyE{2n-1}@{\tfrac{1}{3}} = -\EulerpolyE{2n-1}@{\tfrac{2}{3}}</math>]] || <code>euler(2*n - 1, (1)/(3)) = - euler(2*n - 1, (2)/(3))</code> || <code>EulerE[2*n - 1, Divide[1,3]] == - EulerE[2*n - 1, Divide[2,3]]</code> || Failure || Failure || Successful [Tested: 1] || Successful [Tested: 3]
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| | [https://dlmf.nist.gov/24.4.E30 24.4.E30] || [[Item:Q7443|<math>-\EulerpolyE{2n-1}@{\tfrac{2}{3}} = -\frac{(1-3^{1-2n})(2^{2n}-1)}{2n}\BernoullinumberB{2n}</math>]] || <code>- euler(2*n - 1, (2)/(3)) = -((1 - (3)^(1 - 2*n))*((2)^(2*n)- 1))/(2*n)*bernoulli(2*n)</code> || <code>- EulerE[2*n - 1, Divide[2,3]] == -Divide[(1 - (3)^(1 - 2*n))*((2)^(2*n)- 1),2*n]*BernoulliB[2*n]</code> || Failure || Failure || Successful [Tested: 1] || Successful [Tested: 3]
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| |
| | [https://dlmf.nist.gov/24.4.E31 24.4.E31] || [[Item:Q7444|<math>\BernoullipolyB{n}@{\tfrac{1}{4}} = (-1)^{n}\BernoullipolyB{n}@{\tfrac{3}{4}}</math>]] || <code>bernoulli(n, (1)/(4)) = (- 1)^(n)* bernoulli(n, (3)/(4))</code> || <code>BernoulliB[n, Divide[1,4]] == (- 1)^(n)* BernoulliB[n, Divide[3,4]]</code> || Failure || Successful || Successful [Tested: 1] || Successful [Tested: 1]
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| | [https://dlmf.nist.gov/24.4.E31 24.4.E31] || [[Item:Q7444|<math>(-1)^{n}\BernoullipolyB{n}@{\tfrac{3}{4}} = -\frac{1-2^{1-n}}{2^{n}}\BernoullinumberB{n}-\frac{n}{4^{n}}\EulernumberE{n-1}</math>]] || <code>Error</code> || <code>(- 1)^(n)* BernoulliB[n, Divide[3,4]] == -Divide[1 - (2)^(1 - n),(2)^(n)]*BernoulliB[n]-Divide[n,(4)^(n)]*EulerE[n - 1]</code> || Missing Macro Error || Failure || Skip - symbolical successful subtest || Successful [Tested: 3]
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| |-
| |
| | [https://dlmf.nist.gov/24.4.E32 24.4.E32] || [[Item:Q7445|<math>\BernoullipolyB{2n}@{\tfrac{1}{6}} = \BernoullipolyB{2n}@{\tfrac{5}{6}}</math>]] || <code>bernoulli(2*n, (1)/(6)) = bernoulli(2*n, (5)/(6))</code> || <code>BernoulliB[2*n, Divide[1,6]] == BernoulliB[2*n, Divide[5,6]]</code> || Failure || Failure || Successful [Tested: 3] || Successful [Tested: 3]
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| | [https://dlmf.nist.gov/24.4.E32 24.4.E32] || [[Item:Q7445|<math>\BernoullipolyB{2n}@{\tfrac{5}{6}} = \tfrac{1}{2}(1-2^{1-2n})(1-3^{1-2n})\BernoullinumberB{2n}</math>]] || <code>bernoulli(2*n, (5)/(6)) = (1)/(2)*(1 - (2)^(1 - 2*n))*(1 - (3)^(1 - 2*n))* bernoulli(2*n)</code> || <code>BernoulliB[2*n, Divide[5,6]] == Divide[1,2]*(1 - (2)^(1 - 2*n))*(1 - (3)^(1 - 2*n))* BernoulliB[2*n]</code> || Failure || Failure || Successful [Tested: 3] || Successful [Tested: 3]
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| |
| | [https://dlmf.nist.gov/24.4.E33 24.4.E33] || [[Item:Q7446|<math>\EulerpolyE{2n}@{\tfrac{1}{6}} = \EulerpolyE{2n}@{\tfrac{5}{6}}</math>]] || <code>euler(2*n, (1)/(6)) = euler(2*n, (5)/(6))</code> || <code>EulerE[2*n, Divide[1,6]] == EulerE[2*n, Divide[5,6]]</code> || Failure || Failure || Successful [Tested: 3] || Successful [Tested: 3]
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| | [https://dlmf.nist.gov/24.4.E33 24.4.E33] || [[Item:Q7446|<math>\EulerpolyE{2n}@{\tfrac{5}{6}} = \frac{1+3^{-2n}}{2^{2n+1}}\EulernumberE{2n}</math>]] || <code>Error</code> || <code>EulerE[2*n, Divide[5,6]] == Divide[1 + (3)^(- 2*n),(2)^(2*n + 1)]*EulerE[2*n]</code> || Missing Macro Error || Failure || Skip - symbolical successful subtest || Successful [Tested: 3]
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| |
| | [https://dlmf.nist.gov/24.4.E34 24.4.E34] || [[Item:Q7447|<math>\deriv{}{x}\BernoullipolyB{n}@{x} = n\BernoullipolyB{n-1}@{x}</math>]] || <code>diff(bernoulli(n, x), x) = n*bernoulli(n - 1, x)</code> || <code>D[BernoulliB[n, x], x] == n*BernoulliB[n - 1, x]</code> || Successful || Successful || - || Successful [Tested: 3]
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| |-
| |
| | [https://dlmf.nist.gov/24.4.E35 24.4.E35] || [[Item:Q7448|<math>\deriv{}{x}\EulerpolyE{n}@{x} = n\EulerpolyE{n-1}@{x}</math>]] || <code>diff(euler(n, x), x) = n*euler(n - 1, x)</code> || <code>D[EulerE[n, x], x] == n*EulerE[n - 1, x]</code> || Successful || Successful || - || Successful [Tested: 3]
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| |-
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| | [https://dlmf.nist.gov/24.4.E37 24.4.E37] || [[Item:Q7450|<math>\BernoullipolyB{n}@{x+h} = (B(x)+h)^{n}</math>]] || <code>bernoulli(n, x + h) = (B*(x)+ h)^(n)</code> || <code>BernoulliB[n, x + h] == (B*(x)+ h)^(n)</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><code>300/300]: [[-.299038106-.7500000000*I <- {B = 1/2*3^(1/2)+1/2*I, h = 1/2*3^(1/2)+1/2*I, x = 3/2, n = 1}</code><br><code>.23717473e-1-3.546633371*I <- {B = 1/2*3^(1/2)+1/2*I, h = 1/2*3^(1/2)+1/2*I, x = 3/2, n = 2}</code><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><code>{Complex[-0.299038105676658, -0.7499999999999998] <- {Rule[B, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[h, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[n, 1], Rule[x, 1.5]}</code><br><code>Complex[0.023717474235543268, -3.546633369868303] <- {Rule[B, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[h, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[n, 2], Rule[x, 1.5]}</code><br></div></div>
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| | [https://dlmf.nist.gov/24.4.E38 24.4.E38] || [[Item:Q7451|<math>P(E(x)+1)+P(E(x)) = 2P(x)</math>]] || <code>P*(E*(x)+ 1)+ P*(E*(x)) = 2*P*(x)</code> || <code>P*(E*(x)+ 1)+ P*(E*(x)) == 2*P*(x)</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| | [https://dlmf.nist.gov/24.4.E39 24.4.E39] || [[Item:Q7452|<math>\EulerpolyE{n}@{x+h} = (E(x)+h)^{n}</math>]] || <code>euler(n, x + h) = (E*(x)+ h)^(n)</code> || <code>EulerE[n, x + h] == (E*(x)+ h)^(n)</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><code>300/300]: [[-.299038106-.7500000000*I <- {E = 1/2*3^(1/2)+1/2*I, h = 1/2*3^(1/2)+1/2*I, x = 3/2, n = 1}</code><br><code>-.142949194-3.546633371*I <- {E = 1/2*3^(1/2)+1/2*I, h = 1/2*3^(1/2)+1/2*I, x = 3/2, n = 2}</code><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><code>{Complex[-0.299038105676658, -0.7499999999999998] <- {Rule[E, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[h, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[n, 1], Rule[x, 1.5]}</code><br><code>Complex[-0.14294919243112325, -3.546633369868303] <- {Rule[E, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[h, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[n, 2], Rule[x, 1.5]}</code><br></div></div>
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| | [https://dlmf.nist.gov/24.5.E1 24.5.E1] || [[Item:Q7453|<math>\sum_{k=0}^{n-1}{n\choose k}\BernoullipolyB{k}@{x} = nx^{n-1}</math>]] || <code>sum(binomial(n,k)*bernoulli(k, x), k = 0..n - 1) = n*(x)^(n - 1)</code> || <code>Sum[Binomial[n,k]*BernoulliB[k, x], {k, 0, n - 1}, GenerateConditions->None] == n*(x)^(n - 1)</code> || Failure || Failure || Successful [Tested: 3] || Successful [Tested: 9]
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| | [https://dlmf.nist.gov/24.5.E2 24.5.E2] || [[Item:Q7454|<math>\sum_{k=0}^{n}{n\choose k}\EulerpolyE{k}@{x}+\EulerpolyE{n}@{x} = 2x^{n}</math>]] || <code>sum(binomial(n,k)*euler(k, x), k = 0..n)+ euler(n, x) = 2*(x)^(n)</code> || <code>Sum[Binomial[n,k]*EulerE[k, x], {k, 0, n}, GenerateConditions->None]+ EulerE[n, x] == 2*(x)^(n)</code> || Failure || Failure || Successful [Tested: 3] || Successful [Tested: 9]
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| |-
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| | [https://dlmf.nist.gov/24.5.E3 24.5.E3] || [[Item:Q7455|<math>\sum_{k=0}^{n-1}{n\choose k}\BernoullinumberB{k} = 0</math>]] || <code>sum(binomial(n,k)*bernoulli(k), k = 0..n - 1) = 0</code> || <code>Sum[Binomial[n,k]*BernoulliB[k], {k, 0, n - 1}, GenerateConditions->None] == 0</code> || Failure || Successful || Successful [Tested: 1] || Successful [Tested: 1]
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| |-
| |
| | [https://dlmf.nist.gov/24.5.E4 24.5.E4] || [[Item:Q7456|<math>\sum_{k=0}^{n}{2n\choose 2k}\EulernumberE{2k} = 0</math>]] || <code>Error</code> || <code>Sum[Binomial[2*n,2*k]*EulerE[2*k], {k, 0, n}, GenerateConditions->None] == 0</code> || Missing Macro Error || Failure || - || Successful [Tested: 3]
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| |-
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| | [https://dlmf.nist.gov/24.5.E5 24.5.E5] || [[Item:Q7457|<math>\sum_{k=0}^{n}{n\choose k}2^{k}\EulernumberE{n-k}+\EulernumberE{n} = 2</math>]] || <code>Error</code> || <code>Sum[Binomial[n,k]*(2)^(k)* EulerE[n - k], {k, 0, n}, GenerateConditions->None]+ EulerE[n] == 2</code> || Missing Macro Error || Failure || - || Successful [Tested: 3]
| |
| |-
| |
| | [https://dlmf.nist.gov/24.5.E6 24.5.E6] || [[Item:Q7458|<math>\sum_{k=2}^{n}{n\choose k-2}\frac{\BernoullinumberB{k}}{k} = \frac{1}{(n+1)(n+2)}-\BernoullinumberB{n+1}</math>]] || <code>sum(binomial(n,k - 2)*(bernoulli(k))/(k), k = 2..n) = (1)/((n + 1)*(n + 2))- bernoulli(n + 1)</code> || <code>Sum[Binomial[n,k - 2]*Divide[BernoulliB[k],k], {k, 2, n}, GenerateConditions->None] == Divide[1,(n + 1)*(n + 2)]- BernoulliB[n + 1]</code> || Failure || Failure || Successful [Tested: 1] || Successful [Tested: 3]
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| |-
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| | [https://dlmf.nist.gov/24.5.E7 24.5.E7] || [[Item:Q7459|<math>\sum_{k=0}^{n}{n\choose k}\frac{\BernoullinumberB{k}}{n+2-k} = \frac{\BernoullinumberB{n+1}}{n+1}</math>]] || <code>sum(binomial(n,k)*(bernoulli(k))/(n + 2 - k), k = 0..n) = (bernoulli(n + 1))/(n + 1)</code> || <code>Sum[Binomial[n,k]*Divide[BernoulliB[k],n + 2 - k], {k, 0, n}, GenerateConditions->None] == Divide[BernoulliB[n + 1],n + 1]</code> || Failure || Failure || Successful [Tested: 1] || Successful [Tested: 3]
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| | [https://dlmf.nist.gov/24.5.E8 24.5.E8] || [[Item:Q7460|<math>\sum_{k=0}^{n}\frac{2^{2k}\BernoullinumberB{2k}}{(2k)!(2n+1-2k)!} = \frac{1}{(2n)!}</math>]] || <code>sum(((2)^(2*k)* bernoulli(2*k))/(factorial(2*k)*factorial(2*n + 1 - 2*k)), k = 0..n) = (1)/(factorial(2*n))</code> || <code>Sum[Divide[(2)^(2*k)* BernoulliB[2*k],(2*k)!*(2*n + 1 - 2*k)!], {k, 0, n}, GenerateConditions->None] == Divide[1,(2*n)!]</code> || Failure || Failure || Successful [Tested: 1] || Successful [Tested: 3]
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| |
| | [https://dlmf.nist.gov/24.5#Ex1 24.5#Ex1] || [[Item:Q7461|<math>a_{n} = \sum_{k=0}^{n}{n\choose k}\frac{b_{n-k}}{k+1}</math>]] || <code>a[n] = sum(binomial(n,k)*(b[n - k])/(k + 1), k = 0..n)</code> || <code>Subscript[a, n] == Sum[Binomial[n,k]*Divide[Subscript[b, n - k],k + 1], {k, 0, n}, GenerateConditions->None]</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| | [https://dlmf.nist.gov/24.5#Ex2 24.5#Ex2] || [[Item:Q7462|<math>b_{n} = \sum_{k=0}^{n}{n\choose k}\BernoullinumberB{k}a_{n-k}</math>]] || <code>b[n] = sum(binomial(n,k)*bernoulli(k)*a[n - k], k = 0..n)</code> || <code>Subscript[b, n] == Sum[Binomial[n,k]*BernoulliB[k]*Subscript[a, n - k], {k, 0, n}, GenerateConditions->None]</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><code>300/300]: [[.4330127020+.2500000000*I <- {a[n-k] = 1/2*3^(1/2)+1/2*I, b[n] = 1/2*3^(1/2)+1/2*I, n = 1}</code><br><code>.7216878367+.4166666667*I <- {a[n-k] = 1/2*3^(1/2)+1/2*I, b[n] = 1/2*3^(1/2)+1/2*I, n = 2}</code><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><code>{Complex[0.43301270189221935, 0.24999999999999997] <- {Rule[n, 1], Rule[Subscript[a, Plus[Times[-1, k], n]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[b, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</code><br><code>Complex[0.7216878364870323, 0.41666666666666663] <- {Rule[n, 2], Rule[Subscript[a, Plus[Times[-1, k], n]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[b, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</code><br></div></div>
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| | [https://dlmf.nist.gov/24.5#Ex3 24.5#Ex3] || [[Item:Q7463|<math>a_{n} = \sum_{k=0}^{\floor{\ifrac{n}{2}}}{n\choose 2k}b_{n-2k}</math>]] || <code>a[n] = sum(binomial(n,2*k)*b[n - 2*k], k = 0..floor((n)/(2)))</code> || <code>Subscript[a, n] == Sum[Binomial[n,2*k]*Subscript[b, n - 2*k], {k, 0, Floor[Divide[n,2]]}, GenerateConditions->None]</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [288 / 300]<div class="mw-collapsible-content"><code>288/300]: [[-.8660254040-.5000000000*I <- {a[n] = 1/2*3^(1/2)+1/2*I, b[n-2*k] = 1/2*3^(1/2)+1/2*I, n = 2}</code><br><code>-2.598076212-1.500000000*I <- {a[n] = 1/2*3^(1/2)+1/2*I, b[n-2*k] = 1/2*3^(1/2)+1/2*I, n = 3}</code><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [288 / 300]<div class="mw-collapsible-content"><code>{Complex[-0.8660254037844387, -0.49999999999999994] <- {Rule[n, 2], Rule[Subscript[a, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[b, Plus[Times[-2, k], n]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</code><br><code>Complex[-2.598076211353316, -1.4999999999999998] <- {Rule[n, 3], Rule[Subscript[a, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[b, Plus[Times[-2, k], n]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</code><br></div></div>
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| | [https://dlmf.nist.gov/24.5#Ex4 24.5#Ex4] || [[Item:Q7464|<math>b_{n} = \sum_{k=0}^{\floor{\ifrac{n}{2}}}{n\choose 2k}\EulernumberE{2k}a_{n-2k}</math>]] || <code>Error</code> || <code>Subscript[b, n] == Sum[Binomial[n,2*k]*EulerE[2*k]*Subscript[a, n - 2*k], {k, 0, Floor[Divide[n,2]]}, GenerateConditions->None]</code> || Missing Macro Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [290 / 300]<div class="mw-collapsible-content"><code>{Complex[0.8660254037844387, 0.49999999999999994] <- {Rule[n, 2], Rule[Subscript[a, Plus[Times[-2, k], n]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[b, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</code><br><code>Complex[2.598076211353316, 1.4999999999999998] <- {Rule[n, 3], Rule[Subscript[a, Plus[Times[-2, k], n]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[b, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</code><br></div></div>
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| | [https://dlmf.nist.gov/24.6.E1 24.6.E1] || [[Item:Q7465|<math>\BernoullinumberB{2n} = \sum_{k=2}^{2n+1}\frac{(-1)^{k-1}}{k}{2n+1\choose k}\sum_{j=1}^{k-1}j^{2n}</math>]] || <code>bernoulli(2*n) = sum(((- 1)^(k - 1))/(k)*binomial(2*n + 1,k)*sum((j)^(2*n), j = 1..k - 1), k = 2..2*n + 1)</code> || <code>BernoulliB[2*n] == Sum[Divide[(- 1)^(k - 1),k]*Binomial[2*n + 1,k]*Sum[(j)^(2*n), {j, 1, k - 1}, GenerateConditions->None], {k, 2, 2*n + 1}, GenerateConditions->None]</code> || Failure || Failure || Successful [Tested: 3] || Successful [Tested: 3]
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| | [https://dlmf.nist.gov/24.6.E2 24.6.E2] || [[Item:Q7466|<math>\BernoullinumberB{n} = \frac{1}{n+1}\sum_{k=1}^{n}\sum_{j=1}^{k}(-1)^{j}j^{n}{\binom{n+1}{k-j}}\Bigg{/}{\binom{n}{k}}</math>]] || <code>bernoulli(n) = (1)/(n + 1)*sum(sum((- 1)^(j)* (j)^(n)*binomial(n + 1,k - j)/(binomial(n,k)), j = 1..k), k = 1..n)</code> || <code>BernoulliB[n] == Divide[1,n + 1]*Sum[Sum[(- 1)^(j)* (j)^(n)*Binomial[n + 1,k - j]/(Binomial[n,k]), {j, 1, k}, GenerateConditions->None], {k, 1, n}, GenerateConditions->None]</code> || Aborted || Failure || Successful [Tested: 3] || Successful [Tested: 3]
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| | [https://dlmf.nist.gov/24.6.E3 24.6.E3] || [[Item:Q7467|<math>\BernoullinumberB{2n} = \sum_{k=1}^{n}\frac{(k-1)!k!}{(2k+1)!}\*\sum_{j=1}^{k}(-1)^{j-1}{2k\choose k+j}j^{2n}</math>]] || <code>bernoulli(2*n) = sum((factorial(k - 1)*factorial(k))/(factorial(2*k + 1))* sum((- 1)^(j - 1)*binomial(2*k,k + j)*(j)^(2*n), j = 1..k), k = 1..n)</code> || <code>BernoulliB[2*n] == Sum[Divide[(k - 1)!*(k)!,(2*k + 1)!]* Sum[(- 1)^(j - 1)*Binomial[2*k,k + j]*(j)^(2*n), {j, 1, k}, GenerateConditions->None], {k, 1, n}, GenerateConditions->None]</code> || Aborted || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [3 / 3]<div class="mw-collapsible-content"><code>3/3]: [[.1666666667 <- {n = 1}</code><br><code>-.3333333333e-1 <- {n = 2}</code><br></div></div> || Successful [Tested: 3]
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| | [https://dlmf.nist.gov/24.6.E4 24.6.E4] || [[Item:Q7468|<math>\EulernumberE{2n} = \sum_{k=1}^{n}\frac{1}{2^{k-1}}\sum_{j=1}^{k}(-1)^{j}{2k\choose k-j}j^{2n}</math>]] || <code>Error</code> || <code>EulerE[2*n] == Sum[Divide[1,(2)^(k - 1)]*Sum[(- 1)^(j)*Binomial[2*k,k - j]*(j)^(2*n), {j, 1, k}, GenerateConditions->None], {k, 1, n}, GenerateConditions->None]</code> || Missing Macro Error || Failure || - || Successful [Tested: 3]
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| | [https://dlmf.nist.gov/24.6.E5 24.6.E5] || [[Item:Q7469|<math>\EulernumberE{2n} = \frac{1}{2^{n-1}}\sum_{k=0}^{n-1}(-1)^{n-k}(n-k)^{2n}\*\sum_{j=0}^{k}{2n-2j\choose k-j}2^{j}</math>]] || <code>Error</code> || <code>EulerE[2*n] == Divide[1,(2)^(n - 1)]*Sum[(- 1)^(n - k)*(n - k)^(2*n)* Sum[Binomial[2*n - 2*j,k - j]*(2)^(j), {j, 0, k}, GenerateConditions->None], {k, 0, n - 1}, GenerateConditions->None]</code> || Missing Macro Error || Failure || - || Successful [Tested: 3]
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| | [https://dlmf.nist.gov/24.6.E6 24.6.E6] || [[Item:Q7470|<math>\EulernumberE{2n} = \sum_{k=1}^{2n}\frac{(-1)^{k}}{2^{k-1}}{2n+1\choose k+1}\*\sum_{j=0}^{\floor{\tfrac{1}{2}k-\tfrac{1}{2}}}{k\choose j}(k-2j)^{2n}</math>]] || <code>Error</code> || <code>EulerE[2*n] == Sum[Divide[(- 1)^(k),(2)^(k - 1)]*Binomial[2*n + 1,k + 1]* Sum[Binomial[k,j]*(k - 2*j)^(2*n), {j, 0, Floor[Divide[1,2]*k -Divide[1,2]]}, GenerateConditions->None], {k, 1, 2*n}, GenerateConditions->None]</code> || Missing Macro Error || Failure || - || Successful [Tested: 3]
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| | [https://dlmf.nist.gov/24.6.E7 24.6.E7] || [[Item:Q7471|<math>\BernoullipolyB{n}@{x} = \sum_{k=0}^{n}\frac{1}{k+1}\sum_{j=0}^{k}(-1)^{j}{k\choose j}(x+j)^{n}</math>]] || <code>bernoulli(n, x) = sum((1)/(k + 1)*sum((- 1)^(j)*binomial(k,j)*(x + j)^(n), j = 0..k), k = 0..n)</code> || <code>BernoulliB[n, x] == Sum[Divide[1,k + 1]*Sum[(- 1)^(j)*Binomial[k,j]*(x + j)^(n), {j, 0, k}, GenerateConditions->None], {k, 0, n}, GenerateConditions->None]</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [7 / 9]<div class="mw-collapsible-content"><code>7/9]: [[1. <- {x = 3/2, n = 1}</code><br><code>.9166666667 <- {x = 3/2, n = 2}</code><br></div></div> || Successful [Tested: 9]
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| | [https://dlmf.nist.gov/24.6.E8 24.6.E8] || [[Item:Q7472|<math>\EulerpolyE{n}@{x} = \frac{1}{2^{n}}\sum_{k=1}^{n+1}\sum_{j=0}^{k-1}(-1)^{j}{n+1\choose k}(x+j)^{n}</math>]] || <code>euler(n, x) = (1)/((2)^(n))*sum(sum((- 1)^(j)*binomial(n + 1,k)*(x + j)^(n), j = 0..k - 1), k = 1..n + 1)</code> || <code>EulerE[n, x] == Divide[1,(2)^(n)]*Sum[Sum[(- 1)^(j)*Binomial[n + 1,k]*(x + j)^(n), {j, 0, k - 1}, GenerateConditions->None], {k, 1, n + 1}, GenerateConditions->None]</code> || Failure || Failure || Successful [Tested: 9] || Successful [Tested: 9]
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| | [https://dlmf.nist.gov/24.6.E9 24.6.E9] || [[Item:Q7473|<math>\BernoullinumberB{n} = \sum_{k=0}^{n}\frac{1}{k+1}\sum_{j=0}^{k}(-1)^{j}{k\choose j}j^{n}</math>]] || <code>bernoulli(n) = sum((1)/(k + 1)*sum((- 1)^(j)*binomial(k,j)*(j)^(n), j = 0..k), k = 0..n)</code> || <code>BernoulliB[n] == Sum[Divide[1,k + 1]*Sum[(- 1)^(j)*Binomial[k,j]*(j)^(n), {j, 0, k}, GenerateConditions->None], {k, 0, n}, GenerateConditions->None]</code> || Failure || Successful || <div class="toccolours mw-collapsible mw-collapsed">Failed [2 / 3]<div class="mw-collapsible-content"><code>2/3]: [[-.5000000000 <- {n = 1}</code><br><code>.1666666667 <- {n = 2}</code><br></div></div> || Successful [Tested: 3]
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| | [https://dlmf.nist.gov/24.6.E10 24.6.E10] || [[Item:Q7474|<math>\EulernumberE{n} = \frac{1}{2^{n}}\sum_{k=1}^{n+1}{n+1\choose k}\sum_{j=0}^{k-1}(-1)^{j}(2j+1)^{n}</math>]] || <code>Error</code> || <code>EulerE[n] == Divide[1,(2)^(n)]*Sum[Binomial[n + 1,k]*Sum[(- 1)^(j)*(2*j + 1)^(n), {j, 0, k - 1}, GenerateConditions->None], {k, 1, n + 1}, GenerateConditions->None]</code> || Missing Macro Error || Failure || - || Successful [Tested: 3]
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| | [https://dlmf.nist.gov/24.6.E11 24.6.E11] || [[Item:Q7475|<math>\BernoullinumberB{n} = \frac{n}{2^{n}(2^{n}-1)}\sum_{k=1}^{n}\sum_{j=0}^{k-1}(-1)^{j+1}{n\choose k}j^{n-1}</math>]] || <code>bernoulli(n) = (n)/((2)^(n)*((2)^(n)- 1))*sum(sum((- 1)^(j + 1)*binomial(n,k)*(j)^(n - 1), j = 0..k - 1), k = 1..n)</code> || <code>BernoulliB[n] == Divide[n,(2)^(n)*((2)^(n)- 1)]*Sum[Sum[(- 1)^(j + 1)*Binomial[n,k]*(j)^(n - 1), {j, 0, k - 1}, GenerateConditions->None], {k, 1, n}, GenerateConditions->None]</code> || Aborted || Aborted || Successful [Tested: 3] || Skipped - Because timed out
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| | [https://dlmf.nist.gov/24.6.E12 24.6.E12] || [[Item:Q7476|<math>\EulernumberE{2n} = \sum_{k=0}^{2n}\frac{1}{2^{k}}\sum_{j=0}^{k}(-1)^{j}{k\choose j}(1+2j)^{2n}</math>]] || <code>Error</code> || <code>EulerE[2*n] == Sum[Divide[1,(2)^(k)]*Sum[(- 1)^(j)*Binomial[k,j]*(1 + 2*j)^(2*n), {j, 0, k}, GenerateConditions->None], {k, 0, 2*n}, GenerateConditions->None]</code> || Missing Macro Error || Failure || - || Successful [Tested: 3]
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| | [https://dlmf.nist.gov/24.7.E1 24.7.E1] || [[Item:Q7477|<math>\BernoullinumberB{2n} = (-1)^{n+1}\frac{4n}{1-2^{1-2n}}\int_{0}^{\infty}\frac{t^{2n-1}}{e^{2\pi t}+1}\diff{t}</math>]] || <code>bernoulli(2*n) = (- 1)^(n + 1)*(4*n)/(1 - (2)^(1 - 2*n))*int(((t)^(2*n - 1))/(exp(2*Pi*t)+ 1), t = 0..infinity)</code> || <code>BernoulliB[2*n] == (- 1)^(n + 1)*Divide[4*n,1 - (2)^(1 - 2*n)]*Integrate[Divide[(t)^(2*n - 1),Exp[2*Pi*t]+ 1], {t, 0, Infinity}, GenerateConditions->None]</code> || Failure || Failure || Successful [Tested: 3] || Successful [Tested: 3]
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| | [https://dlmf.nist.gov/24.7.E1 24.7.E1] || [[Item:Q7477|<math>(-1)^{n+1}\frac{4n}{1-2^{1-2n}}\int_{0}^{\infty}\frac{t^{2n-1}}{e^{2\pi t}+1}\diff{t} = (-1)^{n+1}\frac{2n}{1-2^{1-2n}}\int_{0}^{\infty}t^{2n-1}e^{-\pi t}\sech@{\pi t}\diff{t}</math>]] || <code>(- 1)^(n + 1)*(4*n)/(1 - (2)^(1 - 2*n))*int(((t)^(2*n - 1))/(exp(2*Pi*t)+ 1), t = 0..infinity) = (- 1)^(n + 1)*(2*n)/(1 - (2)^(1 - 2*n))*int((t)^(2*n - 1)* exp(- Pi*t)*sech(Pi*t), t = 0..infinity)</code> || <code>(- 1)^(n + 1)*Divide[4*n,1 - (2)^(1 - 2*n)]*Integrate[Divide[(t)^(2*n - 1),Exp[2*Pi*t]+ 1], {t, 0, Infinity}, GenerateConditions->None] == (- 1)^(n + 1)*Divide[2*n,1 - (2)^(1 - 2*n)]*Integrate[(t)^(2*n - 1)* Exp[- Pi*t]*Sech[Pi*t], {t, 0, Infinity}, GenerateConditions->None]</code> || Successful || Successful || Skip - symbolical successful subtest || Successful [Tested: 3]
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| | [https://dlmf.nist.gov/24.7.E2 24.7.E2] || [[Item:Q7478|<math>\BernoullinumberB{2n} = (-1)^{n+1}4n\int_{0}^{\infty}\frac{t^{2n-1}}{e^{2\pi t}-1}\diff{t}</math>]] || <code>bernoulli(2*n) = (- 1)^(n + 1)* 4*n*int(((t)^(2*n - 1))/(exp(2*Pi*t)- 1), t = 0..infinity)</code> || <code>BernoulliB[2*n] == (- 1)^(n + 1)* 4*n*Integrate[Divide[(t)^(2*n - 1),Exp[2*Pi*t]- 1], {t, 0, Infinity}, GenerateConditions->None]</code> || Failure || Failure || Successful [Tested: 3] || Successful [Tested: 3]
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| | [https://dlmf.nist.gov/24.7.E2 24.7.E2] || [[Item:Q7478|<math>(-1)^{n+1}4n\int_{0}^{\infty}\frac{t^{2n-1}}{e^{2\pi t}-1}\diff{t} = (-1)^{n+1}2n\int_{0}^{\infty}t^{2n-1}e^{-\pi t}\csch@{\pi t}\diff{t}</math>]] || <code>(- 1)^(n + 1)* 4*n*int(((t)^(2*n - 1))/(exp(2*Pi*t)- 1), t = 0..infinity) = (- 1)^(n + 1)* 2*n*int((t)^(2*n - 1)* exp(- Pi*t)*csch(Pi*t), t = 0..infinity)</code> || <code>(- 1)^(n + 1)* 4*n*Integrate[Divide[(t)^(2*n - 1),Exp[2*Pi*t]- 1], {t, 0, Infinity}, GenerateConditions->None] == (- 1)^(n + 1)* 2*n*Integrate[(t)^(2*n - 1)* Exp[- Pi*t]*Csch[Pi*t], {t, 0, Infinity}, GenerateConditions->None]</code> || Successful || Successful || Skip - symbolical successful subtest || Successful [Tested: 3]
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| | [https://dlmf.nist.gov/24.7.E3 24.7.E3] || [[Item:Q7479|<math>\BernoullinumberB{2n} = (-1)^{n+1}\frac{\pi}{1-2^{1-2n}}\int_{0}^{\infty}t^{2n}\sech^{2}@{\pi t}\diff{t}</math>]] || <code>bernoulli(2*n) = (- 1)^(n + 1)*(Pi)/(1 - (2)^(1 - 2*n))*int((t)^(2*n)* (sech(Pi*t))^(2), t = 0..infinity)</code> || <code>BernoulliB[2*n] == (- 1)^(n + 1)*Divide[Pi,1 - (2)^(1 - 2*n)]*Integrate[(t)^(2*n)* (Sech[Pi*t])^(2), {t, 0, Infinity}, GenerateConditions->None]</code> || Failure || Aborted || Successful [Tested: 3] || Skipped - Because timed out
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| | [https://dlmf.nist.gov/24.7.E4 24.7.E4] || [[Item:Q7480|<math>\BernoullinumberB{2n} = (-1)^{n+1}\pi\int_{0}^{\infty}t^{2n}\csch^{2}@{\pi t}\diff{t}</math>]] || <code>bernoulli(2*n) = (- 1)^(n + 1)* Pi*int((t)^(2*n)* (csch(Pi*t))^(2), t = 0..infinity)</code> || <code>BernoulliB[2*n] == (- 1)^(n + 1)* Pi*Integrate[(t)^(2*n)* (Csch[Pi*t])^(2), {t, 0, Infinity}, GenerateConditions->None]</code> || Failure || Aborted || Successful [Tested: 3] || Skipped - Because timed out
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| | [https://dlmf.nist.gov/24.7.E5 24.7.E5] || [[Item:Q7481|<math>\BernoullinumberB{2n} = (-1)^{n}\frac{2n(2n-1)}{\pi}\*\int_{0}^{\infty}t^{2n-2}\ln@{1-e^{-2\pi t}}\diff{t}</math>]] || <code>bernoulli(2*n) = (- 1)^(n)*(2*n*(2*n - 1))/(Pi)* int((t)^(2*n - 2)* ln(1 - exp(- 2*Pi*t)), t = 0..infinity)</code> || <code>BernoulliB[2*n] == (- 1)^(n)*Divide[2*n*(2*n - 1),Pi]* Integrate[(t)^(2*n - 2)* Log[1 - Exp[- 2*Pi*t]], {t, 0, Infinity}, GenerateConditions->None]</code> || Failure || Failure || Successful [Tested: 3] || Successful [Tested: 3]
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| | [https://dlmf.nist.gov/24.7.E6 24.7.E6] || [[Item:Q7482|<math>\EulernumberE{2n} = (-1)^{n}2^{2n+1}\int_{0}^{\infty}t^{2n}\sech@{\pi t}\diff{t}</math>]] || <code>Error</code> || <code>EulerE[2*n] == (- 1)^(n)* (2)^(2*n + 1)* Integrate[(t)^(2*n)* Sech[Pi*t], {t, 0, Infinity}, GenerateConditions->None]</code> || Missing Macro Error || Failure || - || Successful [Tested: 3]
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| | [https://dlmf.nist.gov/24.7.E7 24.7.E7] || [[Item:Q7483|<math>\BernoullipolyB{2n}@{x} = (-1)^{n+1}2n\*\int_{0}^{\infty}\frac{\cos@{2\pi x}-e^{-2\pi t}}{\cosh@{2\pi t}-\cos@{2\pi x}}t^{2n-1}\diff{t}</math>]] || <code>bernoulli(2*n, x) = (- 1)^(n + 1)* 2*n * int((cos(2*Pi*x)- exp(- 2*Pi*t))/(cosh(2*Pi*t)- cos(2*Pi*x))*(t)^(2*n - 1), t = 0..infinity)</code> || <code>BernoulliB[2*n, x] == (- 1)^(n + 1)* 2*n * Integrate[Divide[Cos[2*Pi*x]- Exp[- 2*Pi*t],Cosh[2*Pi*t]- Cos[2*Pi*x]]*(t)^(2*n - 1), {t, 0, Infinity}, GenerateConditions->None]</code> || Failure || Aborted || <div class="toccolours mw-collapsible mw-collapsed">Failed [2 / 3]<div class="mw-collapsible-content"><code>2/3]: [[.1875000000 <- {x = 3/2, n = 3}</code><br><code>6.000000000 <- {x = 2, n = 3}</code><br></div></div> || Skipped - Because timed out
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| | [https://dlmf.nist.gov/24.7.E8 24.7.E8] || [[Item:Q7484|<math>\BernoullipolyB{2n+1}@{x} = (-1)^{n+1}(2n+1)\*\int_{0}^{\infty}\frac{\sin@{2\pi x}}{\cosh@{2\pi t}-\cos@{2\pi x}}t^{2n}\diff{t}</math>]] || <code>bernoulli(2*n + 1, x) = (- 1)^(n + 1)*(2*n + 1)* int((sin(2*Pi*x))/(cosh(2*Pi*t)- cos(2*Pi*x))*(t)^(2*n), t = 0..infinity)</code> || <code>BernoulliB[2*n + 1, x] == (- 1)^(n + 1)*(2*n + 1)* Integrate[Divide[Sin[2*Pi*x],Cosh[2*Pi*t]- Cos[2*Pi*x]]*(t)^(2*n), {t, 0, Infinity}, GenerateConditions->None]</code> || Failure || Aborted || <div class="toccolours mw-collapsible mw-collapsed">Failed [6 / 9]<div class="mw-collapsible-content"><code>6/9]: [[.7500000000 <- {x = 3/2, n = 1}</code><br><code>.3125000000 <- {x = 3/2, n = 2}</code><br></div></div> || Skipped - Because timed out
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| | [https://dlmf.nist.gov/24.7.E9 24.7.E9] || [[Item:Q7485|<math>\EulerpolyE{2n}@{x} = (-1)^{n}4\int_{0}^{\infty}\frac{\sin@{\pi x}\cosh@{\pi t}}{\cosh@{2\pi t}-\cos@{2\pi x}}t^{2n}\diff{t}</math>]] || <code>euler(2*n, x) = (- 1)^(n)* 4*int((sin(Pi*x)*cosh(Pi*t))/(cosh(2*Pi*t)- cos(2*Pi*x))*(t)^(2*n), t = 0..infinity)</code> || <code>EulerE[2*n, x] == (- 1)^(n)* 4*Integrate[Divide[Sin[Pi*x]*Cosh[Pi*t],Cosh[2*Pi*t]- Cos[2*Pi*x]]*(t)^(2*n), {t, 0, Infinity}, GenerateConditions->None]</code> || Failure || Aborted || <div class="toccolours mw-collapsible mw-collapsed">Failed [6 / 9]<div class="mw-collapsible-content"><code>6/9]: [[.5000000001 <- {x = 3/2, n = 1}</code><br><code>.1249999998 <- {x = 3/2, n = 2}</code><br></div></div> || Skipped - Because timed out
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| | [https://dlmf.nist.gov/24.7.E10 24.7.E10] || [[Item:Q7486|<math>\EulerpolyE{2n+1}@{x} = (-1)^{n+1}4\*\int_{0}^{\infty}\frac{\cos@{\pi x}\sinh@{\pi t}}{\cosh@{2\pi t}-\cos@{2\pi x}}t^{2n+1}\diff{t}</math>]] || <code>euler(2*n + 1, x) = (- 1)^(n + 1)* 4 * int((cos(Pi*x)*sinh(Pi*t))/(cosh(2*Pi*t)- cos(2*Pi*x))*(t)^(2*n + 1), t = 0..infinity)</code> || <code>EulerE[2*n + 1, x] == (- 1)^(n + 1)* 4 * Integrate[Divide[Cos[Pi*x]*Sinh[Pi*t],Cosh[2*Pi*t]- Cos[2*Pi*x]]*(t)^(2*n + 1), {t, 0, Infinity}, GenerateConditions->None]</code> || Failure || Aborted || <div class="toccolours mw-collapsible mw-collapsed">Failed [6 / 9]<div class="mw-collapsible-content"><code>6/9]: [[.2499999999 <- {x = 3/2, n = 1}</code><br><code>.6250000031e-1 <- {x = 3/2, n = 2}</code><br></div></div> || Skipped - Because timed out
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| | [https://dlmf.nist.gov/24.7.E11 24.7.E11] || [[Item:Q7487|<math>\BernoullipolyB{n}@{x} = \frac{1}{2\pi i}\int_{-c-i\infty}^{-c+i\infty}(x+t)^{n}\left(\frac{\pi}{\sin@{\pi t}}\right)^{2}\diff{t}</math>]] || <code>bernoulli(n, x) = (1)/(2*Pi*I)*int((x + t)^(n)*((Pi)/(sin(Pi*t)))^(2), t = - c - I*infinity..- c + I*infinity)</code> || <code>BernoulliB[n, x] == Divide[1,2*Pi*I]*Integrate[(x + t)^(n)*(Divide[Pi,Sin[Pi*t]])^(2), {t, - c - I*Infinity, - c + I*Infinity}, GenerateConditions->None]</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [7 / 9]<div class="mw-collapsible-content"><code>7/9]: [[1. <- {c = 1/2, x = 3/2, n = 1}</code><br><code>.9166666667 <- {c = 1/2, x = 3/2, n = 2}</code><br></div></div> || Skipped - Because timed out
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| | [https://dlmf.nist.gov/24.8.E1 24.8.E1] || [[Item:Q7488|<math>\BernoullipolyB{2n}@{x} = (-1)^{n+1}\frac{2(2n)!}{(2\pi)^{2n}}\sum_{k=1}^{\infty}\frac{\cos@{2\pi kx}}{k^{2n}}</math>]] || <code>bernoulli(2*n, x) = (- 1)^(n + 1)*(2*factorial(2*n))/((2*Pi)^(2*n))*sum((cos(2*Pi*k*x))/((k)^(2*n)), k = 1..infinity)</code> || <code>BernoulliB[2*n, x] == (- 1)^(n + 1)*Divide[2*(2*n)!,(2*Pi)^(2*n)]*Sum[Divide[Cos[2*Pi*k*x],(k)^(2*n)], {k, 1, Infinity}, GenerateConditions->None]</code> || Aborted || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [6 / 9]<div class="mw-collapsible-content"><code>6/9]: [[1.000000000 <- {x = 3/2, n = 1}</code><br><code>.5000000000 <- {x = 3/2, n = 2}</code><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [6 / 9]<div class="mw-collapsible-content"><code>{Complex[1.0, 0.0] <- {Rule[n, 1], Rule[x, 1.5]}</code><br><code>Complex[0.5000000000000001, 0.0] <- {Rule[n, 2], Rule[x, 1.5]}</code><br></div></div>
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| |-
| |
| | [https://dlmf.nist.gov/24.8.E2 24.8.E2] || [[Item:Q7489|<math>\BernoullipolyB{2n+1}@{x} = (-1)^{n+1}\frac{2(2n+1)!}{(2\pi)^{2n+1}}\sum_{k=1}^{\infty}\frac{\sin@{2\pi kx}}{k^{2n+1}}</math>]] || <code>bernoulli(2*n + 1, x) = (- 1)^(n + 1)*(2*factorial(2*n + 1))/((2*Pi)^(2*n + 1))*sum((sin(2*Pi*k*x))/((k)^(2*n + 1)), k = 1..infinity)</code> || <code>BernoulliB[2*n + 1, x] == (- 1)^(n + 1)*Divide[2*(2*n + 1)!,(2*Pi)^(2*n + 1)]*Sum[Divide[Sin[2*Pi*k*x],(k)^(2*n + 1)], {k, 1, Infinity}, GenerateConditions->None]</code> || Aborted || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [6 / 9]<div class="mw-collapsible-content"><code>{Complex[0.75, 0.0] <- {Rule[n, 1], Rule[x, 1.5]}</code><br><code>Complex[0.3125, 0.0] <- {Rule[n, 2], Rule[x, 1.5]}</code><br></div></div>
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| |
| | [https://dlmf.nist.gov/24.8.E4 24.8.E4] || [[Item:Q7491|<math>\EulerpolyE{2n}@{x} = (-1)^{n}\frac{4(2n)!}{\pi^{2n+1}}\sum_{k=0}^{\infty}\frac{\sin@{(2k+1)\pi x}}{(2k+1)^{2n+1}}</math>]] || <code>euler(2*n, x) = (- 1)^(n)*(4*factorial(2*n))/((Pi)^(2*n + 1))*sum((sin((2*k + 1)* Pi*x))/((2*k + 1)^(2*n + 1)), k = 0..infinity)</code> || <code>EulerE[2*n, x] == (- 1)^(n)*Divide[4*(2*n)!,(Pi)^(2*n + 1)]*Sum[Divide[Sin[(2*k + 1)* Pi*x],(2*k + 1)^(2*n + 1)], {k, 0, Infinity}, GenerateConditions->None]</code> || Aborted || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [6 / 9]<div class="mw-collapsible-content"><code>6/9]: [[.5000000000+0.*I <- {x = 3/2, n = 1}</code><br><code>.1249999998+0.*I <- {x = 3/2, n = 2}</code><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [6 / 9]<div class="mw-collapsible-content"><code>{Complex[0.4999999999999999, -6.717074394942855*^-17] <- {Rule[n, 1], Rule[x, 1.5]}</code><br><code>Complex[0.1250000000000001, -1.3482715791848248*^-17] <- {Rule[n, 2], Rule[x, 1.5]}</code><br></div></div>
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| | [https://dlmf.nist.gov/24.8.E5 24.8.E5] || [[Item:Q7492|<math>\EulerpolyE{2n-1}@{x} = (-1)^{n}\frac{4(2n-1)!}{\pi^{2n}}\sum_{k=0}^{\infty}\frac{\cos@{(2k+1)\pi x}}{(2k+1)^{2n}}</math>]] || <code>euler(2*n - 1, x) = (- 1)^(n)*(4*factorial(2*n - 1))/((Pi)^(2*n))*sum((cos((2*k + 1)* Pi*x))/((2*k + 1)^(2*n)), k = 0..infinity)</code> || <code>EulerE[2*n - 1, x] == (- 1)^(n)*Divide[4*(2*n - 1)!,(Pi)^(2*n)]*Sum[Divide[Cos[(2*k + 1)* Pi*x],(2*k + 1)^(2*n)], {k, 0, Infinity}, GenerateConditions->None]</code> || Aborted || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [6 / 9]<div class="mw-collapsible-content"><code>6/9]: [[1. <- {x = 3/2, n = 1}</code><br><code>.2500000000 <- {x = 3/2, n = 2}</code><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [6 / 9]<div class="mw-collapsible-content"><code>{Complex[1.0, -2.3810929344395102*^-33] <- {Rule[n, 1], Rule[x, 1.5]}</code><br><code>Complex[0.25000000000000006, 5.146963577016199*^-33] <- {Rule[n, 2], Rule[x, 1.5]}</code><br></div></div>
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| | [https://dlmf.nist.gov/24.8.E6 24.8.E6] || [[Item:Q7493|<math>\BernoullinumberB{4n+2} = (8n+4)\sum_{k=1}^{\infty}\frac{k^{4n+1}}{e^{2\pi k}-1}</math>]] || <code>bernoulli(4*n + 2) = (8*n + 4)* sum(((k)^(4*n + 1))/(exp(2*Pi*k)- 1), k = 1..infinity)</code> || <code>BernoulliB[4*n + 2] == (8*n + 4)* Sum[Divide[(k)^(4*n + 1),Exp[2*Pi*k]- 1], {k, 1, Infinity}, GenerateConditions->None]</code> || Failure || Failure || Successful [Tested: 1] || <div class="toccolours mw-collapsible mw-collapsed">Failed [3 / 3]<div class="mw-collapsible-content"><code>{Plus[0.023809523809523808, Times[-12.0, NSum[Times[Power[Plus[-1, Power[E, Times[2, k, Pi]]], -1], Power[k, 5]] <- {k, 1, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[n, 1]}</code><br><code>Plus[0.07575757575757576, Times[-20.0, NSum[Times[Power[Plus[-1, Power[E, Times[2, k, Pi]]], -1], Power[k, 9]] <- {k, 1, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[n, 2]}</code><br></div></div>
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| |-
| |
| | [https://dlmf.nist.gov/24.8.E7 24.8.E7] || [[Item:Q7494|<math>\BernoullinumberB{2n} = \frac{(-1)^{n+1}4n}{2^{2n}-1}\sum_{k=1}^{\infty}\frac{k^{2n-1}}{e^{\pi k}+(-1)^{k+n}}</math>]] || <code>bernoulli(2*n) = ((- 1)^(n + 1)* 4*n)/((2)^(2*n)- 1)*sum(((k)^(2*n - 1))/(exp(Pi*k)+(- 1)^(k + n)), k = 1..infinity)</code> || <code>BernoulliB[2*n] == Divide[(- 1)^(n + 1)* 4*n,(2)^(2*n)- 1]*Sum[Divide[(k)^(2*n - 1),Exp[Pi*k]+(- 1)^(k + n)], {k, 1, Infinity}, GenerateConditions->None]</code> || Failure || Failure || Successful [Tested: 1] || <div class="toccolours mw-collapsible mw-collapsed">Failed [3 / 3]<div class="mw-collapsible-content"><code>{Plus[0.16666666666666666, Times[-1.3333333333333333, NSum[Times[Power[Plus[Power[-1, Plus[1, k]], Power[E, Times[k, Pi]]], -1], k] <- {k, 1, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[n, 1]}</code><br><code>Plus[-0.03333333333333333, Times[0.5333333333333333, NSum[Times[Power[Plus[Power[-1, Plus[2, k]], Power[E, Times[k, Pi]]], -1], Power[k, 3]] <- {k, 1, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[n, 2]}</code><br></div></div>
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| |-
| |
| | [https://dlmf.nist.gov/24.8.E8 24.8.E8] || [[Item:Q7495|<math>\frac{\BernoullinumberB{2n}}{4n}\left(\alpha^{n}-(-\beta)^{n}\right) = \alpha^{n}\sum_{k=1}^{\infty}\frac{k^{2n-1}}{e^{2\alpha k}-1}-(-\beta)^{n}\sum_{k=1}^{\infty}\frac{k^{2n-1}}{e^{2\beta k}-1}</math>]] || <code>(bernoulli(2*n))/(4*n)*((alpha)^(n)-(- beta)^(n)) = (alpha)^(n)* sum(((k)^(2*n - 1))/(exp(2*alpha*k)- 1), k = 1..infinity)-(- beta)^(n)* sum(((k)^(2*n - 1))/(exp(2*beta*k)- 1), k = 1..infinity)</code> || <code>Divide[BernoulliB[2*n],4*n]*(\[Alpha]^(n)-(- \[Beta])^(n)) == \[Alpha]^(n)* Sum[Divide[(k)^(2*n - 1),Exp[2*\[Alpha]*k]- 1], {k, 1, Infinity}, GenerateConditions->None]-(- \[Beta])^(n)* Sum[Divide[(k)^(2*n - 1),Exp[2*\[Beta]*k]- 1], {k, 1, Infinity}, GenerateConditions->None]</code> || Failure || Aborted || <div class="toccolours mw-collapsible mw-collapsed">Failed [6 / 9]<div class="mw-collapsible-content"><code>6/9]: [[1.443014212 <- {alpha = 3/2, beta = 1/2, n = 2}</code><br><code>-.7774267192e-1 <- {alpha = 3/2, beta = 2, n = 2}</code><br></div></div> || Skipped - Because timed out
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| | [https://dlmf.nist.gov/24.8.E9 24.8.E9] || [[Item:Q7496|<math>{}\EulernumberE{2n} = (-1)^{n}\sum_{k=1}^{\infty}\frac{k^{2n}}{\cosh@{\tfrac{1}{2}\pi k}}-4\sum_{k=0}^{\infty}\frac{(-1)^{k}(2k+1)^{2n}}{e^{2\pi(2k+1)}-1}</math>]] || <code>Error</code> || <code>null*EulerE[2*n] == (- 1)^(n)* Sum[Divide[(k)^(2*n),Cosh[Divide[1,2]*Pi*k]], {k, 1, Infinity}, GenerateConditions->None]- 4*Sum[Divide[(- 1)^(k)*(2*k + 1)^(2*n),Exp[2*Pi*(2*k + 1)]- 1], {k, 0, Infinity}, GenerateConditions->None]</code> || Translation Error || Translation Error || - || -
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| | [https://dlmf.nist.gov/24.9.E1 24.9.E1] || [[Item:Q7497|<math>|\BernoullinumberB{2n}| > |\BernoullipolyB{2n}@{x}|</math>]] || <code>abs(bernoulli(2*n)) > abs(bernoulli(2*n, x))</code> || <code>Abs[BernoulliB[2*n]] > Abs[BernoulliB[2*n, x]]</code> || Failure || Failure || Successful [Tested: 3] || Successful [Tested: 3]
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| | [https://dlmf.nist.gov/24.9.E2 24.9.E2] || [[Item:Q7498|<math>(2-2^{1-2n})|\BernoullinumberB{2n}| \geq |\BernoullipolyB{2n}@{x}-\BernoullinumberB{2n}|</math>]] || <code>(2 - (2)^(1 - 2*n))*abs(bernoulli(2*n)) >= abs(bernoulli(2*n, x)- bernoulli(2*n))</code> || <code>(2 - (2)^(1 - 2*n))*Abs[BernoulliB[2*n]] >= Abs[BernoulliB[2*n, x]- BernoulliB[2*n]]</code> || Failure || Failure || Successful [Tested: 3] || Successful [Tested: 3]
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| | [https://dlmf.nist.gov/24.9.E3 24.9.E3] || [[Item:Q7499|<math>4^{-n}|\EulernumberE{2n}| > (-1)^{n}\EulerpolyE{2n}@{x}</math>]] || <code>Error</code> || <code>(4)^(- n)*Abs[EulerE[2*n]] > (- 1)^(n)* EulerE[2*n, x]</code> || Missing Macro Error || Failure || Skip - symbolical successful subtest || <div class="toccolours mw-collapsible mw-collapsed">Failed [4 / 9]<div class="mw-collapsible-content"><code>{False <- {Rule[n, 1], Rule[x, 0.5]}</code><br><code>False <- {Rule[n, 2], Rule[x, 0.5]}</code><br></div></div>
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| | [https://dlmf.nist.gov/24.9.E3 24.9.E3] || [[Item:Q7499|<math>(-1)^{n}\EulerpolyE{2n}@{x} > 0</math>]] || <code>(- 1)^(n)* euler(2*n, x) > 0</code> || <code>(- 1)^(n)* EulerE[2*n, x] > 0</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [5 / 9]<div class="mw-collapsible-content"><code>5/9]: [[0. < -.7500000000 <- {x = 3/2, n = 1}</code><br><code>0. < -.1875000000 <- {x = 3/2, n = 2}</code><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [5 / 9]<div class="mw-collapsible-content"><code>{False <- {Rule[n, 1], Rule[x, 1.5]}</code><br><code>False <- {Rule[n, 2], Rule[x, 1.5]}</code><br></div></div>
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| | [https://dlmf.nist.gov/24.9.E4 24.9.E4] || [[Item:Q7500|<math>\frac{2(2n+1)!}{(2\pi)^{2n+1}} > (-1)^{n+1}\BernoullipolyB{2n+1}@{x}</math>]] || <code>(2*factorial(2*n + 1))/((2*Pi)^(2*n + 1)) > (- 1)^(n + 1)* bernoulli(2*n + 1, x)</code> || <code>Divide[2*(2*n + 1)!,(2*Pi)^(2*n + 1)] > (- 1)^(n + 1)* BernoulliB[2*n + 1, x]</code> || Failure || Failure || Successful [Tested: 3] || <div class="toccolours mw-collapsible mw-collapsed">Failed [4 / 9]<div class="mw-collapsible-content"><code>{False <- {Rule[n, 1], Rule[x, 1.5]}</code><br><code>False <- {Rule[n, 3], Rule[x, 1.5]}</code><br></div></div>
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| | [https://dlmf.nist.gov/24.9.E4 24.9.E4] || [[Item:Q7500|<math>(-1)^{n+1}\BernoullipolyB{2n+1}@{x} > 0</math>]] || <code>(- 1)^(n + 1)* bernoulli(2*n + 1, x) > 0</code> || <code>(- 1)^(n + 1)* BernoulliB[2*n + 1, x] > 0</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [3 / 3]<div class="mw-collapsible-content"><code>3/3]: [[0. < -.3125000000 <- {x = 3/2, n = 2}</code><br><code>0. < 0. <- {x = 1/2, n = 2}</code><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [5 / 9]<div class="mw-collapsible-content"><code>{False <- {Rule[n, 2], Rule[x, 1.5]}</code><br><code>False <- {Rule[n, 1], Rule[x, 0.5]}</code><br></div></div>
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| | [https://dlmf.nist.gov/24.9.E5 24.9.E5] || [[Item:Q7501|<math>\frac{4(2n-1)!}{\pi^{2n}}\frac{2^{2n}-1}{2^{2n}-2} > (-1)^{n}\EulerpolyE{2n-1}@{x}</math>]] || <code>(4*factorial(2*n - 1))/((Pi)^(2*n))*((2)^(2*n)- 1)/((2)^(2*n)- 2) > (- 1)^(n)* euler(2*n - 1, x)</code> || <code>Divide[4*(2*n - 1)!,(Pi)^(2*n)]*Divide[(2)^(2*n)- 1,(2)^(2*n)- 2] > (- 1)^(n)* EulerE[2*n - 1, x]</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [1 / 9]<div class="mw-collapsible-content"><code>1/9]: [[2.250000000 < .2639824007 <- {x = 2, n = 2}</code><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [1 / 9]<div class="mw-collapsible-content"><code>{False <- {Rule[n, 2], Rule[x, 2]}</code><br></div></div>
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| | [https://dlmf.nist.gov/24.9.E5 24.9.E5] || [[Item:Q7501|<math>(-1)^{n}\EulerpolyE{2n-1}@{x} > 0</math>]] || <code>(- 1)^(n)* euler(2*n - 1, x) > 0</code> || <code>(- 1)^(n)* EulerE[2*n - 1, x] > 0</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [7 / 9]<div class="mw-collapsible-content"><code>7/9]: [[0. < -1. <- {x = 3/2, n = 1}</code><br><code>0. < -.6250000000e-1 <- {x = 3/2, n = 3}</code><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [7 / 9]<div class="mw-collapsible-content"><code>{False <- {Rule[n, 1], Rule[x, 1.5]}</code><br><code>False <- {Rule[n, 3], Rule[x, 1.5]}</code><br></div></div>
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| | [https://dlmf.nist.gov/24.9.E6 24.9.E6] || [[Item:Q7502|<math>5\sqrt{\pi n}\left(\frac{n}{\pi e}\right)^{2n} > (-1)^{n+1}\BernoullinumberB{2n}</math>]] || <code>5*sqrt(Pi*n)*((n)/(Pi*exp(1)))^(2*n) > (- 1)^(n + 1)* bernoulli(2*n)</code> || <code>5*Sqrt[Pi*n]*(Divide[n,Pi*E])^(2*n) > (- 1)^(n + 1)* BernoulliB[2*n]</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [1 / 3]<div class="mw-collapsible-content"><code>1/3]: [[.1666666667 < .1215223702 <- {n = 1}</code><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [1 / 3]<div class="mw-collapsible-content"><code>{False <- {Rule[n, 1]}</code><br></div></div>
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| | [https://dlmf.nist.gov/24.9.E6 24.9.E6] || [[Item:Q7502|<math>(-1)^{n+1}\BernoullinumberB{2n} > 4\sqrt{\pi n}\left(\frac{n}{\pi e}\right)^{2n}</math>]] || <code>(- 1)^(n + 1)* bernoulli(2*n) > 4*sqrt(Pi*n)*((n)/(Pi*exp(1)))^(2*n)</code> || <code>(- 1)^(n + 1)* BernoulliB[2*n] > 4*Sqrt[Pi*n]*(Divide[n,Pi*E])^(2*n)</code> || Failure || Failure || Successful [Tested: 3] || Successful [Tested: 3]
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| | [https://dlmf.nist.gov/24.9.E7 24.9.E7] || [[Item:Q7503|<math>8\sqrt{\frac{n}{\pi}}\left(\frac{4n}{\pi e}\right)^{2n}\left(1+\frac{1}{12n}\right) > (-1)^{n}\EulernumberE{2n}</math>]] || <code>Error</code> || <code>8*Sqrt[Divide[n,Pi]]*(Divide[4*n,Pi*E])^(2*n)*(1 +Divide[1,12*n]) > (- 1)^(n)* EulerE[2*n]</code> || Missing Macro Error || Failure || - || Successful [Tested: 3]
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| |-
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| | [https://dlmf.nist.gov/24.9.E7 24.9.E7] || [[Item:Q7503|<math>(-1)^{n}\EulernumberE{2n} > 8\sqrt{\frac{n}{\pi}}\left(\frac{4n}{\pi e}\right)^{2n}</math>]] || <code>Error</code> || <code>(- 1)^(n)* EulerE[2*n] > 8*Sqrt[Divide[n,Pi]]*(Divide[4*n,Pi*E])^(2*n)</code> || Missing Macro Error || Failure || - || Successful [Tested: 3]
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| | [https://dlmf.nist.gov/24.9.E8 24.9.E8] || [[Item:Q7504|<math>\frac{2(2n)!}{(2\pi)^{2n}}\frac{1}{1-2^{\beta-2n}} \geq (-1)^{n+1}\BernoullinumberB{2n}\geq\frac{2(2n)!}{(2\pi)^{2n}}\frac{1}{1-2^{-2n}}</math>]] || <code>(2*factorial(2*n))/((2*Pi)^(2*n))*(1)/(1 - (2)^(beta - 2*n)) >= (- 1)^(n + 1)* bernoulli(2*n) >= (2*factorial(2*n))/((2*Pi)^(2*n))*(1)/(1 - (2)^(- 2*n))</code> || <code>Divide[2*(2*n)!,(2*Pi)^(2*n)]*Divide[1,1 - (2)^(\[Beta]- 2*n)] >= (- 1)^(n + 1)* BernoulliB[2*n] >= Divide[2*(2*n)!,(2*Pi)^(2*n)]*Divide[1,1 - (2)^(- 2*n)]</code> || Failure || Failure || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [2 / 9]<div class="mw-collapsible-content"><code>{False <- {Rule[n, 1], Rule[β, 0.5]}</code><br><code>GreaterEqual[DirectedInfinity[], 0.16666666666666666] <- {Rule[n, 1], Rule[β, 2]}</code><br></div></div>
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| | [https://dlmf.nist.gov/24.9.E9 24.9.E9] || [[Item:Q7505|<math>\beta = 2+\frac{\ln@{1-6\pi^{-2}}}{\ln@@{2}}</math>]] || <code>beta = 2 +(ln(1 - 6*(Pi)^(- 2)))/(ln(2))</code> || <code>\[Beta] == 2 +Divide[Log[1 - 6*(Pi)^(- 2)],Log[2]]</code> || Aborted || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [3 / 3]<div class="mw-collapsible-content"><code>3/3]: [[.850806174 <- {beta = 3/2}</code><br><code>-.1491938260 <- {beta = 1/2}</code><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [3 / 3]<div class="mw-collapsible-content"><code>{0.850806175200028 <- {Rule[β, 1.5]}</code><br><code>-0.149193824799972 <- {Rule[β, 0.5]}</code><br></div></div>
| |
| |-
| |
| | [https://dlmf.nist.gov/24.9.E9 24.9.E9] || [[Item:Q7505|<math>2+\frac{\ln@{1-6\pi^{-2}}}{\ln@@{2}} = 0.6491\dots</math>]] || <code>2 +(ln(1 - 6*(Pi)^(- 2)))/(ln(2)) = 0.6491</code> || <code>2 +Divide[Log[1 - 6*(Pi)^(- 2)],Log[2]] == 0.6491</code> || Error || Failure || Skip - symbolical successful subtest || Successful [Tested: 1]
| |
| |-
| |
| | [https://dlmf.nist.gov/24.9.E10 24.9.E10] || [[Item:Q7506|<math>\frac{4^{n+1}(2n)!}{\pi^{2n+1}} > (-1)^{n}\EulernumberE{2n}</math>]] || <code>Error</code> || <code>Divide[(4)^(n + 1)*(2*n)!,(Pi)^(2*n + 1)] > (- 1)^(n)* EulerE[2*n]</code> || Missing Macro Error || Failure || - || Successful [Tested: 3]
| |
| |-
| |
| | [https://dlmf.nist.gov/24.9.E10 24.9.E10] || [[Item:Q7506|<math>(-1)^{n}\EulernumberE{2n} > \frac{4^{n+1}(2n)!}{\pi^{2n+1}}\frac{1}{1+3^{-1-2n}}</math>]] || <code>Error</code> || <code>(- 1)^(n)* EulerE[2*n] > Divide[(4)^(n + 1)*(2*n)!,(Pi)^(2*n + 1)]*Divide[1,1 + (3)^(- 1 - 2*n)]</code> || Missing Macro Error || Failure || - || Successful [Tested: 3]
| |
| |-
| |
| | [https://dlmf.nist.gov/24.12.E1 24.12.E1] || [[Item:Q7522|<math>\tfrac{1}{2} \leq x_{1}^{(n)}</math>]] || <code>(1)/(2) <= (x[1])^(n)</code> || <code>Divide[1,2] <= (Subscript[x, 1])^(n)</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
| |
| |-
| |
| | [https://dlmf.nist.gov/24.12.E2 24.12.E2] || [[Item:Q7523|<math>\frac{3}{4}+\frac{1}{2^{n+2}\pi} < x^{(n)}_{1}</math>]] || <code>(3)/(4)+(1)/((2)^(n + 2)* Pi) < (x[1])^(n)</code> || <code>Divide[3,4]+Divide[1,(2)^(n + 2)* Pi] < (Subscript[x, 1])^(n)</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
| |
| |-
| |
| | [https://dlmf.nist.gov/24.12.E7 24.12.E7] || [[Item:Q7530|<math>\tfrac{1}{2} \leq y^{(n)}_{1}</math>]] || <code>(1)/(2) <= (y[1])^(n)</code> || <code>Divide[1,2] <= (Subscript[y, 1])^(n)</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
| |
| |-
| |
| | [https://dlmf.nist.gov/24.12.E9 24.12.E9] || [[Item:Q7532|<math>\frac{3}{2}-\frac{\pi^{n+1}}{3(n!)} < y^{(n)}_{2}</math>]] || <code>(3)/(2)-((Pi)^(n + 1))/(3*(factorial(n))) < (y[2])^(n)</code> || <code>Divide[3,2]-Divide[(Pi)^(n + 1),3*((n)!)] < (Subscript[y, 2])^(n)</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
| |
| |-
| |
| | [https://dlmf.nist.gov/24.12.E10 24.12.E10] || [[Item:Q7533|<math>\frac{3}{2} < y^{(n)}_{2}</math>]] || <code>(3)/(2) < (y[2])^(n)</code> || <code>Divide[3,2] < (Subscript[y, 2])^(n)</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
| |
| |-
| |
| | [https://dlmf.nist.gov/24.13.E2 24.13.E2] || [[Item:Q7536|<math>\int_{x}^{x+1}\BernoullipolyB{n}@{t}\diff{t} = x^{n}</math>]] || <code>int(bernoulli(n, t), t = x..x + 1) = (x)^(n)</code> || <code>Integrate[BernoulliB[n, t], {t, x, x + 1}, GenerateConditions->None] == (x)^(n)</code> || Failure || Failure || Successful [Tested: 3] || Successful [Tested: 9]
| |
| |-
| |
| | [https://dlmf.nist.gov/24.13.E3 24.13.E3] || [[Item:Q7537|<math>\int_{x}^{x+(1/2)}\BernoullipolyB{n}@{t}\diff{t} = \frac{\EulerpolyE{n}@{2x}}{2^{n+1}}</math>]] || <code>int(bernoulli(n, t), t = x..x +(1/ 2)) = (euler(n, 2*x))/((2)^(n + 1))</code> || <code>Integrate[BernoulliB[n, t], {t, x, x +(1/ 2)}, GenerateConditions->None] == Divide[EulerE[n, 2*x],(2)^(n + 1)]</code> || Failure || Failure || Successful [Tested: 9] || Successful [Tested: 9]
| |
| |-
| |
| | [https://dlmf.nist.gov/24.13.E4 24.13.E4] || [[Item:Q7538|<math>\int_{0}^{1/2}\BernoullipolyB{n}@{t}\diff{t} = \frac{1-2^{n+1}}{2^{n}}\frac{\BernoullinumberB{n+1}}{n+1}</math>]] || <code>int(bernoulli(n, t), t = 0..1/ 2) = (1 - (2)^(n + 1))/((2)^(n))*(bernoulli(n + 1))/(n + 1)</code> || <code>Integrate[BernoulliB[n, t], {t, 0, 1/ 2}, GenerateConditions->None] == Divide[1 - (2)^(n + 1),(2)^(n)]*Divide[BernoulliB[n + 1],n + 1]</code> || Failure || Failure || Successful [Tested: 3] || Successful [Tested: 3]
| |
| |-
| |
| | [https://dlmf.nist.gov/24.13.E5 24.13.E5] || [[Item:Q7539|<math>\int_{1/4}^{3/4}\BernoullipolyB{n}@{t}\diff{t} = \frac{\EulernumberE{n}}{2^{2n+1}}</math>]] || <code>Error</code> || <code>Integrate[BernoulliB[n, t], {t, 1/ 4, 3/ 4}, GenerateConditions->None] == Divide[EulerE[n],(2)^(2*n + 1)]</code> || Missing Macro Error || Failure || - || Successful [Tested: 3]
| |
| |-
| |
| | [https://dlmf.nist.gov/24.13.E6 24.13.E6] || [[Item:Q7540|<math>\int_{0}^{1}\BernoullipolyB{n}@{t}\BernoullipolyB{m}@{t}\diff{t} = \frac{(-1)^{n-1}m!n!}{(m+n)!}\BernoullinumberB{m+n}</math>]] || <code>int(bernoulli(n, t)*bernoulli(m, t), t = 0..1) = ((- 1)^(n - 1)* factorial(m)*factorial(n))/(factorial(m + n))*bernoulli(m + n)</code> || <code>Integrate[BernoulliB[n, t]*BernoulliB[m, t], {t, 0, 1}, GenerateConditions->None] == Divide[(- 1)^(n - 1)* (m)!*(n)!,(m + n)!]*BernoulliB[m + n]</code> || Failure || Failure || Successful [Tested: 9] || Successful [Tested: 9]
| |
| |-
| |
| | [https://dlmf.nist.gov/24.13.E8 24.13.E8] || [[Item:Q7542|<math>\int_{0}^{1}\EulerpolyE{n}@{t}\diff{t} = -2\frac{\EulerpolyE{n+1}@{0}}{n+1}</math>]] || <code>int(euler(n, t), t = 0..1) = - 2*(euler(n + 1, 0))/(n + 1)</code> || <code>Integrate[EulerE[n, t], {t, 0, 1}, GenerateConditions->None] == - 2*Divide[EulerE[n + 1, 0],n + 1]</code> || Failure || Failure || Successful [Tested: 3] || Successful [Tested: 3]
| |
| |-
| |
| | [https://dlmf.nist.gov/24.13.E8 24.13.E8] || [[Item:Q7542|<math>-2\frac{\EulerpolyE{n+1}@{0}}{n+1} = \frac{4(2^{n+2}-1)}{(n+1)(n+2)}\BernoullinumberB{n+2}</math>]] || <code>- 2*(euler(n + 1, 0))/(n + 1) = (4*((2)^(n + 2)- 1))/((n + 1)*(n + 2))*bernoulli(n + 2)</code> || <code>- 2*Divide[EulerE[n + 1, 0],n + 1] == Divide[4*((2)^(n + 2)- 1),(n + 1)*(n + 2)]*BernoulliB[n + 2]</code> || Failure || Failure || Successful [Tested: 3] || Successful [Tested: 3]
| |
| |-
| |
| | [https://dlmf.nist.gov/24.13.E9 24.13.E9] || [[Item:Q7543|<math>\int_{0}^{1/2}\EulerpolyE{2n}@{t}\diff{t} = -\frac{\EulerpolyE{2n+1}@{0}}{2n+1}</math>]] || <code>int(euler(2*n, t), t = 0..1/ 2) = -(euler(2*n + 1, 0))/(2*n + 1)</code> || <code>Integrate[EulerE[2*n, t], {t, 0, 1/ 2}, GenerateConditions->None] == -Divide[EulerE[2*n + 1, 0],2*n + 1]</code> || Failure || Failure || Successful [Tested: 3] || Successful [Tested: 3]
| |
| |-
| |
| | [https://dlmf.nist.gov/24.13.E9 24.13.E9] || [[Item:Q7543|<math>-\frac{\EulerpolyE{2n+1}@{0}}{2n+1} = \frac{2(2^{2n+2}-1)\BernoullinumberB{2n+2}}{(2n+1)(2n+2)}</math>]] || <code>-(euler(2*n + 1, 0))/(2*n + 1) = (2*((2)^(2*n + 2)- 1)* bernoulli(2*n + 2))/((2*n + 1)*(2*n + 2))</code> || <code>-Divide[EulerE[2*n + 1, 0],2*n + 1] == Divide[2*((2)^(2*n + 2)- 1)* BernoulliB[2*n + 2],(2*n + 1)*(2*n + 2)]</code> || Failure || Failure || Successful [Tested: 3] || Successful [Tested: 3]
| |
| |-
| |
| | [https://dlmf.nist.gov/24.13.E10 24.13.E10] || [[Item:Q7544|<math>\int_{0}^{1/2}\EulerpolyE{2n-1}@{t}\diff{t} = \frac{\EulernumberE{2n}}{n2^{2n+1}}</math>]] || <code>Error</code> || <code>Integrate[EulerE[2*n - 1, t], {t, 0, 1/ 2}, GenerateConditions->None] == Divide[EulerE[2*n],n*(2)^(2*n + 1)]</code> || Missing Macro Error || Failure || - || Skipped - Because timed out
| |
| |-
| |
| | [https://dlmf.nist.gov/24.13.E11 24.13.E11] || [[Item:Q7545|<math>\int_{0}^{1}\EulerpolyE{n}@{t}\EulerpolyE{m}@{t}\diff{t} = (-1)^{n}4\frac{(2^{m+n+2}-1)m!n!}{(m+n+2)!}\BernoullinumberB{m+n+2}</math>]] || <code>int(euler(n, t)*euler(m, t), t = 0..1) = (- 1)^(n)* 4*(((2)^(m + n + 2)- 1)* factorial(m)*factorial(n))/(factorial(m + n + 2))*bernoulli(m + n + 2)</code> || <code>Integrate[EulerE[n, t]*EulerE[m, t], {t, 0, 1}, GenerateConditions->None] == (- 1)^(n)* 4*Divide[((2)^(m + n + 2)- 1)* (m)!*(n)!,(m + n + 2)!]*BernoulliB[m + n + 2]</code> || Failure || Failure || Successful [Tested: 9] || Successful [Tested: 9]
| |
| |-
| |
| | [https://dlmf.nist.gov/24.14.E1 24.14.E1] || [[Item:Q7546|<math>\sum_{k=0}^{n}{n\choose k}\BernoullipolyB{k}@{x}\BernoullipolyB{n-k}@{y} = n(x+y-1)\BernoullipolyB{n-1}@{x+y}-(n-1)\BernoullipolyB{n}@{x+y}</math>]] || <code>sum(binomial(n,k)*bernoulli(k, x)*bernoulli(n - k, y), k = 0..n) = n*(x + y - 1)* bernoulli(n - 1, x + y)-(n - 1)* bernoulli(n, x + y)</code> || <code>Sum[Binomial[n,k]*BernoulliB[k, x]*BernoulliB[n - k, y], {k, 0, n}, GenerateConditions->None] == n*(x + y - 1)* BernoulliB[n - 1, x + y]-(n - 1)* BernoulliB[n, x + y]</code> || Failure || Successful || Successful [Tested: 54] || Successful [Tested: 54]
| |
| |-
| |
| | [https://dlmf.nist.gov/24.14.E2 24.14.E2] || [[Item:Q7547|<math>\sum_{k=0}^{n}{n\choose k}\BernoullinumberB{k}\BernoullinumberB{n-k} = (1-n)\BernoullinumberB{n}-n\BernoullinumberB{n-1}</math>]] || <code>sum(binomial(n,k)*bernoulli(k)*bernoulli(n - k), k = 0..n) = (1 - n)* bernoulli(n)- n*bernoulli(n - 1)</code> || <code>Sum[Binomial[n,k]*BernoulliB[k]*BernoulliB[n - k], {k, 0, n}, GenerateConditions->None] == (1 - n)* BernoulliB[n]- n*BernoulliB[n - 1]</code> || Failure || Failure || Successful [Tested: 3] || Successful [Tested: 3]
| |
| |-
| |
| | [https://dlmf.nist.gov/24.14.E3 24.14.E3] || [[Item:Q7548|<math>\sum_{k=0}^{n}{n\choose k}\EulerpolyE{k}@{h}\EulerpolyE{n-k}@{x} = 2(\EulerpolyE{n+1}@{x+h}-(x+h-1)\EulerpolyE{n}@{x+h})</math>]] || <code>sum(binomial(n,k)*euler(k, h)*euler(n - k, x), k = 0..n) = 2*(euler(n + 1, x + h)-(x + h - 1)*euler(n, x + h))</code> || <code>Sum[Binomial[n,k]*EulerE[k, h]*EulerE[n - k, x], {k, 0, n}, GenerateConditions->None] == 2*(EulerE[n + 1, x + h]-(x + h - 1)*EulerE[n, x + h])</code> || Failure || Successful || Successful [Tested: 90] || Successful [Tested: 90]
| |
| |-
| |
| | [https://dlmf.nist.gov/24.14.E4 24.14.E4] || [[Item:Q7549|<math>\sum_{k=0}^{n}{n\choose k}\EulernumberE{k}\EulernumberE{n-k} = -2^{n+1}\EulerpolyE{n+1}@{0}</math>]] || <code>Error</code> || <code>Sum[Binomial[n,k]*EulerE[k]*EulerE[n - k], {k, 0, n}, GenerateConditions->None] == - (2)^(n + 1)* EulerE[n + 1, 0]</code> || Missing Macro Error || Failure || Skip - symbolical successful subtest || Successful [Tested: 3]
| |
| |-
| |
| | [https://dlmf.nist.gov/24.14.E4 24.14.E4] || [[Item:Q7549|<math>-2^{n+1}\EulerpolyE{n+1}@{0} = -2^{n+2}(1-2^{n+2})\frac{\BernoullinumberB{n+2}}{n+2}</math>]] || <code>- (2)^(n + 1)* euler(n + 1, 0) = - (2)^(n + 2)*(1 - (2)^(n + 2))*(bernoulli(n + 2))/(n + 2)</code> || <code>- (2)^(n + 1)* EulerE[n + 1, 0] == - (2)^(n + 2)*(1 - (2)^(n + 2))*Divide[BernoulliB[n + 2],n + 2]</code> || Failure || Failure || Successful [Tested: 3] || Successful [Tested: 3]
| |
| |-
| |
| | [https://dlmf.nist.gov/24.14.E5 24.14.E5] || [[Item:Q7550|<math>\sum_{k=0}^{n}{n\choose k}\EulerpolyE{k}@{h}\BernoullipolyB{n-k}@{x} = 2^{n}\BernoullipolyB{n}@{\tfrac{1}{2}(x+h)}</math>]] || <code>sum(binomial(n,k)*euler(k, h)*bernoulli(n - k, x), k = 0..n) = (2)^(n)* bernoulli(n, (1)/(2)*(x + h))</code> || <code>Sum[Binomial[n,k]*EulerE[k, h]*BernoulliB[n - k, x], {k, 0, n}, GenerateConditions->None] == (2)^(n)* BernoulliB[n, Divide[1,2]*(x + h)]</code> || Failure || Failure || Successful [Tested: 90] || Successful [Tested: 90]
| |
| |-
| |
| | [https://dlmf.nist.gov/24.14.E6 24.14.E6] || [[Item:Q7551|<math>\sum_{k=0}^{n}{n\choose k}2^{k}\BernoullinumberB{k}\EulernumberE{n-k} = 2(1-2^{n-1})\BernoullinumberB{n}-n\EulernumberE{n-1}</math>]] || <code>Error</code> || <code>Sum[Binomial[n,k]*(2)^(k)* BernoulliB[k]*EulerE[n - k], {k, 0, n}, GenerateConditions->None] == 2*(1 - (2)^(n - 1))* BernoulliB[n]- n*EulerE[n - 1]</code> || Missing Macro Error || Failure || - || Successful [Tested: 3]
| |
| |-
| |
| | [https://dlmf.nist.gov/24.14.E7 24.14.E7] || [[Item:Q7552|<math>\sum_{j=0}^{m}\sum_{k=0}^{n}\binom{m}{j}\binom{n}{k}\frac{\BernoullinumberB{j}\BernoullinumberB{k}}{m+n-j-k+1} = (-1)^{m-1}\frac{m!n!}{(m+n)!}\BernoullinumberB{m+n}</math>]] || <code>sum(sum(binomial(m,j)*binomial(n,k)*(bernoulli(j)*bernoulli(k))/(m + n - j - k + 1), k = 0..n), j = 0..m) = (- 1)^(m - 1)*(factorial(m)*factorial(n))/(factorial(m + n))*bernoulli(m + n)</code> || <code>Sum[Sum[Binomial[m,j]*Binomial[n,k]*Divide[BernoulliB[j]*BernoulliB[k],m + n - j - k + 1], {k, 0, n}, GenerateConditions->None], {j, 0, m}, GenerateConditions->None] == (- 1)^(m - 1)*Divide[(m)!*(n)!,(m + n)!]*BernoulliB[m + n]</code> || Failure || Failure || Successful [Tested: 9] || Successful [Tested: 9]
| |
| |-
| |
| | [https://dlmf.nist.gov/24.15.E1 24.15.E1] || [[Item:Q7558|<math>\frac{2t}{e^{t}+1} = \sum_{n=1}^{\infty}G_{n}\frac{t^{n}}{n!}</math>]] || <code>(2*t)/(exp(t)+ 1) = sum(G[n]*((t)^(n))/(factorial(n)), n = 1..infinity)</code> || <code>Divide[2*t,Exp[t]+ 1] == Sum[Subscript[G, n]*Divide[(t)^(n),(n)!], {n, 1, Infinity}, GenerateConditions->None]</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
| |
| |-
| |
| | [https://dlmf.nist.gov/24.15.E2 24.15.E2] || [[Item:Q7559|<math>G_{n} = 2(1-2^{n})\BernoullinumberB{n}</math>]] || <code>G[n] = 2*(1 - (2)^(n))* bernoulli(n)</code> || <code>Subscript[G, n] == 2*(1 - (2)^(n))* BernoulliB[n]</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [30 / 30]<div class="mw-collapsible-content"><code>30/30]: [[-.1339745960+.5000000000*I <- {G[n] = 1/2*3^(1/2)+1/2*I, n = 1}</code><br><code>1.866025404+.5000000000*I <- {G[n] = 1/2*3^(1/2)+1/2*I, n = 2}</code><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [30 / 30]<div class="mw-collapsible-content"><code>{Complex[-0.1339745962155613, 0.49999999999999994] <- {Rule[n, 1], Rule[Subscript[G, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</code><br><code>Complex[1.8660254037844388, 0.49999999999999994] <- {Rule[n, 2], Rule[Subscript[G, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</code><br></div></div>
| |
| |-
| |
| | [https://dlmf.nist.gov/24.15.E3 24.15.E3] || [[Item:Q7560|<math>\tan@@{t} = \sum_{n=0}^{\infty}T_{n}\frac{t^{n}}{n!}</math>]] || <code>tan(t) = sum(T[n]*((t)^(n))/(factorial(n)), n = 0..infinity)</code> || <code>Tan[t] == Sum[Subscript[T, n]*Divide[(t)^(n),(n)!], {n, 0, Infinity}, GenerateConditions->None]</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [60 / 60]<div class="mw-collapsible-content"><code>60/60]: [[-14.29465634-.1115650801*I <- {t = -3/2, T[n] = 1/2*3^(1/2)+1/2*I}</code><br><code>-13.98985487-.1932363871*I <- {t = -3/2, T[n] = -1/2+1/2*I*3^(1/2)}</code><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [60 / 60]<div class="mw-collapsible-content"><code>{Complex[-14.29465633421075, -0.1115650800742149] <- {Rule[t, -1.5], Rule[Subscript[T, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</code><br><code>Complex[-13.989854867097504, -0.1932363870390304] <- {Rule[t, -1.5], Rule[Subscript[T, n], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</code><br></div></div>
| |
| |-
| |
| | [https://dlmf.nist.gov/24.15.E4 24.15.E4] || [[Item:Q7561|<math>T_{2n-1} = (-1)^{n-1}\frac{2^{2n}(2^{2n}-1)}{2n}\BernoullinumberB{2n}</math>]] || <code>T[2*n - 1] = (- 1)^(n - 1)*((2)^(2*n)*((2)^(2*n)- 1))/(2*n)*bernoulli(2*n)</code> || <code>Subscript[T, 2*n - 1] == (- 1)^(n - 1)*Divide[(2)^(2*n)*((2)^(2*n)- 1),2*n]*BernoulliB[2*n]</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [10 / 10]<div class="mw-collapsible-content"><code>10/10]: [[-15.13397460+.5000000000*I <- {T[2*n-1] = 1/2*3^(1/2)+1/2*I, n = 3}</code><br><code>-16.50000000+.8660254040*I <- {T[2*n-1] = -1/2+1/2*I*3^(1/2), n = 3}</code><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [29 / 30]<div class="mw-collapsible-content"><code>{Complex[-0.1339745962155613, 0.49999999999999994] <- {Rule[n, 1], Rule[Subscript[T, Plus[-1, Times[2, n]]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</code><br><code>Complex[-1.1339745962155612, 0.49999999999999994] <- {Rule[n, 2], Rule[Subscript[T, Plus[-1, Times[2, n]]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</code><br></div></div>
| |
| |-
| |
| | [https://dlmf.nist.gov/24.15.E5 24.15.E5] || [[Item:Q7562|<math>T_{2n} = 0</math>]] || <code>T[2*n] = 0</code> || <code>Subscript[T, 2*n] == 0</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
| |
| |-
| |
| | [https://dlmf.nist.gov/24.15.E6 24.15.E6] || [[Item:Q7563|<math>\BernoullinumberB{n} = \sum_{k=0}^{n}(-1)^{k}\frac{k!\StirlingnumberS@{n}{k}}{k+1}</math>]] || <code>bernoulli(n) = sum((- 1)^(k)*(factorial(k)*Stirling2(n, k))/(k + 1), k = 0..n)</code> || <code>BernoulliB[n] == Sum[(- 1)^(k)*Divide[(k)!*StirlingS2[n, k],k + 1], {k, 0, n}, GenerateConditions->None]</code> || Failure || Successful || Successful [Tested: 3] || Successful [Tested: 3]
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| | [https://dlmf.nist.gov/24.15.E7 24.15.E7] || [[Item:Q7564|<math>\BernoullinumberB{n} = \sum_{k=0}^{n}(-1)^{k}\binom{n+1}{k+1}\StirlingnumberS@{n+k}{k}\bigg{/}\binom{n+k}{k}</math>]] || <code>bernoulli(n) = sum((- 1)^(k)*binomial(n + 1,k + 1)*Stirling2(n + k, k)/(binomial(n + k,k)), k = 0..n)</code> || <code>BernoulliB[n] == Sum[(- 1)^(k)*Binomial[n + 1,k + 1]*StirlingS2[n + k, k]/(Binomial[n + k,k]), {k, 0, n}, GenerateConditions->None]</code> || Failure || Failure || Successful [Tested: 3] || Successful [Tested: 3]
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| |
| | [https://dlmf.nist.gov/24.15.E8 24.15.E8] || [[Item:Q7565|<math>\sum_{k=0}^{n}(-1)^{n+k}\Stirlingnumbers@{n+1}{k+1}\BernoullinumberB{k} = \frac{n!}{n+1}</math>]] || <code>sum((- 1)^(n + k)* Stirling1(n + 1, k + 1)*bernoulli(k), k = 0..n) = (factorial(n))/(n + 1)</code> || <code>Sum[(- 1)^(n + k)* StirlingS1[n + 1, k + 1]*BernoulliB[k], {k, 0, n}, GenerateConditions->None] == Divide[(n)!,n + 1]</code> || Failure || Failure || Successful [Tested: 3] || Successful [Tested: 3]
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| |
| | [https://dlmf.nist.gov/24.15.E11 24.15.E11] || [[Item:Q7568|<math>\sum_{k=0}^{\floor{\ifrac{n}{2}}}{n\choose 2k}\left(\frac{5}{9}\right)^{k}\BernoullinumberB{2k}u_{n-2k} = \frac{n}{6}v_{n-1}+\frac{n}{3^{n}}v_{2n-2}</math>]] || <code>sum(binomial(n,2*k)*((5)/(9))^(k)* bernoulli(2*k)*u[n - 2*k], k = 0..floor((n)/(2))) = (n)/(6)*v[n - 1]+(n)/((3)^(n))*v[2*n - 2]</code> || <code>Sum[Binomial[n,2*k]*(Divide[5,9])^(k)* BernoulliB[2*k]*Subscript[u, n - 2*k], {k, 0, Floor[Divide[n,2]]}, GenerateConditions->None] == Divide[n,6]*Subscript[v, n - 1]+Divide[n,(3)^(n)]*Subscript[v, 2*n - 2]</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><code>300/300]: [[.4330127020+.2500000000*I <- {u[n-2*k] = 1/2*3^(1/2)+1/2*I, v[n-1] = 1/2*3^(1/2)+1/2*I, v[2*n-2] = 1/2*3^(1/2)+1/2*I, n = 1}</code><br><code>.4650877169+.2685185185*I <- {u[n-2*k] = 1/2*3^(1/2)+1/2*I, v[n-1] = 1/2*3^(1/2)+1/2*I, v[2*n-2] = 1/2*3^(1/2)+1/2*I, n = 2}</code><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><code>{Complex[0.43301270189221935, 0.24999999999999997] <- {Rule[n, 1], Rule[Subscript[u, Plus[Times[-2, k], n]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[v, Plus[-1, n]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[v, Plus[-2, Times[2, n]]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</code><br><code>Complex[0.46508771684719863, 0.2685185185185185] <- {Rule[n, 2], Rule[Subscript[u, Plus[Times[-2, k], n]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[v, Plus[-1, n]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[v, Plus[-2, Times[2, n]]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</code><br></div></div>
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| | [https://dlmf.nist.gov/24.15.E12 24.15.E12] || [[Item:Q7569|<math>\sum_{k=0}^{\floor{\ifrac{n}{2}}}{n\choose 2k}\left(\frac{5}{4}\right)^{k}\EulernumberE{2k}v_{n-2k} = \frac{1}{2^{n-1}}</math>]] || <code>Error</code> || <code>Sum[Binomial[n,2*k]*(Divide[5,4])^(k)* EulerE[2*k]*Subscript[v, n - 2*k], {k, 0, Floor[Divide[n,2]]}, GenerateConditions->None] == Divide[1,(2)^(n - 1)]</code> || Missing Macro Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [29 / 30]<div class="mw-collapsible-content"><code>{Complex[-0.1339745962155613, 0.49999999999999994] <- {Rule[n, 1], Rule[Subscript[v, Plus[Times[-2, k], n]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</code><br><code>Complex[-0.7165063509461097, -0.12499999999999999] <- {Rule[n, 2], Rule[Subscript[v, Plus[Times[-2, k], n]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</code><br></div></div>
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| | [https://dlmf.nist.gov/24.16.E5 24.16.E5] || [[Item:Q7575|<math>\frac{t}{\ln@{1+t}} = \sum_{n=0}^{\infty}b_{n}t^{n}</math>]] || <code>(t)/(ln(1 + t)) = sum(b[n]*(t)^(n), n = 0..infinity)</code> || <code>Divide[t,Log[1 + t]] == Sum[Subscript[b, n]*(t)^(n), {n, 0, Infinity}, GenerateConditions->None]</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [20 / 20]<div class="mw-collapsible-content"><code>20/20]: [[.1439972511-.3333333333*I <- {t = -1/2, b[n] = 1/2*3^(1/2)+1/2*I}</code><br><code>1.054680854-.5773502693*I <- {t = -1/2, b[n] = -1/2+1/2*I*3^(1/2)}</code><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [20 / 20]<div class="mw-collapsible-content"><code>{Complex[0.14399725125485596, -0.33333333333333326] <- {Rule[t, -0.5], Rule[Subscript[b, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</code><br><code>Complex[1.054680853777815, -0.5773502691896257] <- {Rule[t, -0.5], Rule[Subscript[b, n], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</code><br></div></div>
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| | [https://dlmf.nist.gov/24.16.E7 24.16.E7] || [[Item:Q7577|<math>\frac{t}{(1+\lambda t)^{\ifrac{1}{\lambda}}-1} = \sum_{n=0}^{\infty}\beta_{n}(\lambda)\frac{t^{n}}{n!}</math>]] || <code>(t)/((1 + lambda*t)^((1)/(lambda))- 1) = sum(beta[n]*(lambda)*((t)^(n))/(factorial(n)), n = 0..infinity)</code> || <code>Divide[t,(1 + \[Lambda]*t)^(Divide[1,\[Lambda]])- 1] == Sum[Subscript[\[Beta], n]*(\[Lambda])*Divide[(t)^(n),(n)!], {n, 0, Infinity}, GenerateConditions->None]</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| |
| | [https://dlmf.nist.gov/24.16.E8 24.16.E8] || [[Item:Q7578|<math>\beta_{n}(\lambda) = n!b_{n}\lambda^{n}+\sum_{k=1}^{\floor{\ifrac{n}{2}}}\frac{n}{2k}\BernoullinumberB{2k}\Stirlingnumbers@{n-1}{2k-1}\lambda^{n-2k}</math>]] || <code>beta[n]*(lambda) = factorial(n)*b[n]*(lambda)^(n)+ sum((n)/(2*k)*bernoulli(2*k)*Stirling1(n - 1, 2*k - 1)*(lambda)^(n - 2*k), k = 1..floor((n)/(2)))</code> || <code>Subscript[\[Beta], n]*(\[Lambda]) == (n)!*Subscript[b, n]*\[Lambda]^(n)+ Sum[Divide[n,2*k]*BernoulliB[2*k]*StirlingS1[n - 1, 2*k - 1]*\[Lambda]^(n - 2*k), {k, 1, Floor[Divide[n,2]]}, GenerateConditions->None]</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><code>300/300]: [[.3333333331-1.133974598*I <- {beta = 3/2, lambda = 1/2*3^(1/2)+1/2*I, b[n] = 1/2*3^(1/2)+1/2*I, beta[n] = 1/2*3^(1/2)+1/2*I, n = 2}</code><br><code>-1.032692071-1.500000002*I <- {beta = 3/2, lambda = 1/2*3^(1/2)+1/2*I, b[n] = 1/2*3^(1/2)+1/2*I, beta[n] = -1/2+1/2*I*3^(1/2), n = 2}</code><br></div></div> || Skipped - Because timed out
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| | [https://dlmf.nist.gov/24.16.E10 24.16.E10] || [[Item:Q7580|<math>\sum_{a=1}^{f}\frac{\chi(a)te^{at}}{e^{ft}-1} = \sum_{n=0}^{\infty}B_{n,\chi}\frac{t^{n}}{n!}</math>]] || <code>sum((chi*(a)* t*exp(a*t))/(exp(f*t)- 1), a = 1..f) = sum(B[n , chi]*((t)^(n))/(factorial(n)), n = 0..infinity)</code> || <code>Sum[Divide[\[Chi]*(a)* t*Exp[a*t],Exp[f*t]- 1], {a, 1, f}, GenerateConditions->None] == Sum[Subscript[B, n , \[Chi]]*Divide[(t)^(n),(n)!], {n, 0, Infinity}, GenerateConditions->None]</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| |
| | [https://dlmf.nist.gov/24.16.E11 24.16.E11] || [[Item:Q7581|<math>B_{n,\chi}(x) = \sum_{k=0}^{n}{n\choose k}B_{k,\chi}x^{n-k}</math>]] || <code>B[n , chi]*(x) = sum(binomial(n,k)*B[k , chi]*(x)^(n - k), k = 0..n)</code> || <code>Subscript[B, n , \[Chi]]*(x) == Sum[Binomial[n,k]*Subscript[B, k , \[Chi]]*(x)^(n - k), {k, 0, n}, GenerateConditions->None]</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| | [https://dlmf.nist.gov/24.16.E12 24.16.E12] || [[Item:Q7582|<math>\BernoullipolyB{n}@{x} = B_{n,\chi_{0}}(x-1)</math>]] || <code>bernoulli(n, x) = B[n , chi[0]]*(x - 1)</code> || <code>BernoulliB[n, x] == Subscript[B, n , Subscript[\[Chi], 0]]*(x - 1)</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [280 / 300]<div class="mw-collapsible-content"><code>280/300]: [[.5669872980-.2500000000*I <- {chi = 1/2*3^(1/2)+1/2*I, x = 3/2, B[n,chi[0]] = 1/2*3^(1/2)+1/2*I, chi[0] = 1/2*3^(1/2)+1/2*I, n = 1}</code><br><code>.4836539647-.2500000000*I <- {chi = 1/2*3^(1/2)+1/2*I, x = 3/2, B[n,chi[0]] = 1/2*3^(1/2)+1/2*I, chi[0] = 1/2*3^(1/2)+1/2*I, n = 2}</code><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [280 / 300]<div class="mw-collapsible-content"><code>{Complex[0.5669872981077806, -0.24999999999999997] <- {Rule[n, 1], Rule[x, 1.5], Rule[χ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[χ, 0], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[B, n, Subscript[χ, 0]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</code><br><code>Complex[0.4836539647744474, -0.24999999999999997] <- {Rule[n, 2], Rule[x, 1.5], Rule[χ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[χ, 0], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[B, n, Subscript[χ, 0]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</code><br></div></div>
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| | [https://dlmf.nist.gov/24.16.E13 24.16.E13] || [[Item:Q7583|<math>\EulerpolyE{n}@{x} = -\frac{2^{1-n}}{n+1}B_{n+1,\chi_{4}}(2x-1)</math>]] || <code>euler(n, x) = -((2)^(1 - n))/(n + 1)*B[n + 1 , chi[4]]*(2*x - 1)</code> || <code>EulerE[n, x] == -Divide[(2)^(1 - n),n + 1]*Subscript[B, n + 1 , Subscript[\[Chi], 4]]*(2*x - 1)</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [290 / 300]<div class="mw-collapsible-content"><code>290/300]: [[1.866025404+.5000000000*I <- {chi = 1/2*3^(1/2)+1/2*I, x = 3/2, B[n+1,chi[4]] = 1/2*3^(1/2)+1/2*I, chi[4] = 1/2*3^(1/2)+1/2*I, n = 1}</code><br><code>1.038675135+.1666666667*I <- {chi = 1/2*3^(1/2)+1/2*I, x = 3/2, B[n+1,chi[4]] = 1/2*3^(1/2)+1/2*I, chi[4] = 1/2*3^(1/2)+1/2*I, n = 2}</code><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [290 / 300]<div class="mw-collapsible-content"><code>{Complex[1.8660254037844388, 0.49999999999999994] <- {Rule[n, 1], Rule[x, 1.5], Rule[χ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[χ, 4], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[B, Plus[1, n], Subscript[χ, 4]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</code><br><code>Complex[1.0386751345948129, 0.16666666666666663] <- {Rule[n, 2], Rule[x, 1.5], Rule[χ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[χ, 4], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[B, Plus[1, n], Subscript[χ, 4]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</code><br></div></div>
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| | [https://dlmf.nist.gov/24.17.E4 24.17.E4] || [[Item:Q7587|<math>S_{n}(k) = (-1)^{k}</math>]] || <code>S[n]*(k) = (- 1)^(k)</code> || <code>Subscript[S, n]*(k) == (- 1)^(k)</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| | [https://dlmf.nist.gov/24.17.E6 24.17.E6] || [[Item:Q7589|<math>M_{n}(k) = 0</math>]] || <code>M[n]*(k) = 0</code> || <code>Subscript[M, n]*(k) == 0</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| | [https://dlmf.nist.gov/24.19#Ex2 24.19#Ex2] || [[Item:Q7594|<math>\BernoullinumberB{2n} = \dfrac{N_{2n}}{D_{2n}}</math>]] || <code>bernoulli(2*n) = (N[2*n])/(D[2*n])</code> || <code>BernoulliB[2*n] == Divide[Subscript[N, 2*n],Subscript[D, 2*n]]</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><code>300/300]: [[-.8333333333 <- {D[2*n] = 1/2*3^(1/2)+1/2*I, N[2*n] = 1/2*3^(1/2)+1/2*I, n = 1}</code><br><code>-1.033333333 <- {D[2*n] = 1/2*3^(1/2)+1/2*I, N[2*n] = 1/2*3^(1/2)+1/2*I, n = 2}</code><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><code>{-0.8333333333333334 <- {Rule[n, 1], Rule[Subscript[D, Times[2, n]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[N, Times[2, n]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</code><br><code>-1.0333333333333334 <- {Rule[n, 2], Rule[Subscript[D, Times[2, n]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[N, Times[2, n]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</code><br></div></div>
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| | [https://dlmf.nist.gov/24.19.E3 24.19.E3] || [[Item:Q7595|<math>\frac{t^{2}}{\cosh@@{t}-1} = -2\sum_{n=0}^{\infty}(2n-1)\BernoullinumberB{2n}\frac{t^{2n}}{(2n)!}</math>]] || <code>((t)^(2))/(cosh(t)- 1) = - 2*sum((2*n - 1)* bernoulli(2*n)*((t)^(2*n))/(factorial(2*n)), n = 0..infinity)</code> || <code>Divide[(t)^(2),Cosh[t]- 1] == - 2*Sum[(2*n - 1)* BernoulliB[2*n]*Divide[(t)^(2*n),(2*n)!], {n, 0, Infinity}, GenerateConditions->None]</code> || Failure || Aborted || Successful [Tested: 6] || Skipped - Because timed out
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| |}
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