Results of Combinatorial Analysis: Difference between revisions

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Notation: 26.1 Special Notation
Properties: 26.2 Basic Definitions
26.3 Lattice Paths: Binomial Coefficients
26.4 Lattice Paths: Multinomial Coefficients and Set Partitions
26.5 Lattice Paths: Catalan Numbers
26.6 Other Lattice Path Numbers
26.7 Set Partitions: Bell Numbers
26.8 Set Partitions: Stirling Numbers
26.9 Integer Partitions: Restricted Number and Part Size
26.10 Integer Partitions: Other Restrictions
26.11 Integer Partitions: Compositions
26.12 Plane Partitions
26.13 Permutations: Cycle Notation
26.14 Permutations: Order Notation
26.15 Permutations: Matrix Notation
26.16 Multiset Permutations
26.17 The Twelvefold Way
26.18 Counting Techniques
Applications: 26.19 Mathematical Applications
26.20 Physical Applications
Computation: 26.21 Tables
26.22 Software

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+<div style="-moz-column-count:2; column-count:2;">
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+; Notation : [[26.1|26.1 Special Notation]]<br>
-! DLMF !! Formula !! Constraints !! Maple !! Mathematica !! Symbolic<br>Maple !! Symbolic<br>Mathematica !! Numeric<br>Maple !! Numeric<br>Mathematica
+; Properties : [[26.2|26.2 Basic Definitions]]<br>[[26.3|26.3 Lattice Paths: Binomial Coefficients]]<br>[[26.4|26.4 Lattice Paths: Multinomial Coefficients and Set Partitions]]<br>[[26.5|26.5 Lattice Paths: Catalan Numbers]]<br>[[26.6|26.6 Other Lattice Path Numbers]]<br>[[26.7|26.7 Set Partitions: Bell Numbers]]<br>[[26.8|26.8 Set Partitions: Stirling Numbers]]<br>[[26.9|26.9 Integer Partitions:
-|-
+Restricted Number and Part Size]]<br>[[26.10|26.10 Integer Partitions: Other Restrictions]]<br>[[26.11|26.11 Integer Partitions: Compositions]]<br>[[26.12|26.12 Plane Partitions]]<br>[[26.13|26.13 Permutations: Cycle Notation]]<br>[[26.14|26.14 Permutations: Order Notation]]<br>[[26.15|26.15 Permutations: Matrix Notation]]<br>[[26.16|26.16 Multiset Permutations]]<br>[[26.17|26.17 The Twelvefold Way]]<br>[[26.18|26.18 Counting Techniques]]<br>
-| [https://dlmf.nist.gov/26.3.E1 26.3.E1] || [[Item:Q7774|<math>\binom{m}{n} = \binom{m}{m-n}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\binom{m}{n} = \binom{m}{m-n}</syntaxhighlight> || <math>m \geq n</math> || <syntaxhighlight lang=mathematica>binomial(m,n) = binomial(m,m - n)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Binomial[m,n] == Binomial[m,m - n]</syntaxhighlight> || Failure || Successful || Successful [Tested: 6] || Successful [Tested: 6]
+; Applications : [[26.19|26.19 Mathematical Applications]]<br>[[26.20|26.20 Physical Applications]]<br>
-|-
+; Computation : [[26.21|26.21 Tables]]<br>[[26.22|26.22 Software]]<br>
-| [https://dlmf.nist.gov/26.3.E1 26.3.E1] || [[Item:Q7774|<math>\binom{m}{m-n} = \frac{m!}{(m-n)!\,n!}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\binom{m}{m-n} = \frac{m!}{(m-n)!\,n!}</syntaxhighlight> || <math>m \geq n</math> || <syntaxhighlight lang=mathematica>binomial(m,m - n) = (factorial(m))/(factorial(m - n)*factorial(n))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Binomial[m,m - n] == Divide[(m)!,(m - n)!*(n)!]</syntaxhighlight> || Successful || Successful || Skip - symbolical successful subtest || Successful [Tested: 6]
+</div>
-|-
-| [https://dlmf.nist.gov/26.3.E2 26.3.E2] || [[Item:Q7775|<math>\binom{m}{n} = 0</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\binom{m}{n} = 0</syntaxhighlight> || <math>n > m</math> || <syntaxhighlight lang=mathematica>binomial(m,n) = 0</syntaxhighlight> || <syntaxhighlight lang=mathematica>Binomial[m,n] == 0</syntaxhighlight> || Failure || Failure || Successful [Tested: 3] || Successful [Tested: 3]
-|-
-| [https://dlmf.nist.gov/26.3.E3 26.3.E3] || [[Item:Q7776|<math>\sum_{n=0}^{m}\binom{m}{n}x^{n} = (1+x)^{m}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sum_{n=0}^{m}\binom{m}{n}x^{n} = (1+x)^{m}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>sum(binomial(m,n)*(x)^(n), n = 0..m) = (1 + x)^(m)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sum[Binomial[m,n]*(x)^(n), {n, 0, m}, GenerateConditions->None] == (1 + x)^(m)</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 0]
-|-
-| [https://dlmf.nist.gov/26.3.E4 26.3.E4] || [[Item:Q7777|<math>\sum_{m=0}^{\infty}\binom{m+n}{m}x^{m} = \frac{1}{(1-x)^{n+1}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sum_{m=0}^{\infty}\binom{m+n}{m}x^{m} = \frac{1}{(1-x)^{n+1}}</syntaxhighlight> || <math>|x| < 1</math> || <syntaxhighlight lang=mathematica>sum(binomial(m + n,m)*(x)^(m), m = 0..infinity) = (1)/((1 - x)^(n + 1))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sum[Binomial[m + n,m]*(x)^(m), {m, 0, Infinity}, GenerateConditions->None] == Divide[1,(1 - x)^(n + 1)]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 3]
-|-
-| [https://dlmf.nist.gov/26.3.E5 26.3.E5] || [[Item:Q7778|<math>\binom{m}{n} = \binom{m-1}{n}+\binom{m-1}{n-1}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\binom{m}{n} = \binom{m-1}{n}+\binom{m-1}{n-1}</syntaxhighlight> || <math>m \geq n, n \geq 1</math> || <syntaxhighlight lang=mathematica>binomial(m,n) = binomial(m - 1,n)+binomial(m - 1,n - 1)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Binomial[m,n] == Binomial[m - 1,n]+Binomial[m - 1,n - 1]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 6]
-|-
-| [https://dlmf.nist.gov/26.3.E6 26.3.E6] || [[Item:Q7779|<math>\binom{m}{n} = \frac{m}{n}\binom{m-1}{n-1}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\binom{m}{n} = \frac{m}{n}\binom{m-1}{n-1}</syntaxhighlight> || <math>m \geq n, n \geq 1</math> || <syntaxhighlight lang=mathematica>binomial(m,n) = (m)/(n)*binomial(m - 1,n - 1)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Binomial[m,n] == Divide[m,n]*Binomial[m - 1,n - 1]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 6]
-|-
-| [https://dlmf.nist.gov/26.3.E6 26.3.E6] || [[Item:Q7779|<math>\frac{m}{n}\binom{m-1}{n-1} = \frac{m-n+1}{n}\binom{m}{n-1}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\frac{m}{n}\binom{m-1}{n-1} = \frac{m-n+1}{n}\binom{m}{n-1}</syntaxhighlight> || <math>m \geq n, n \geq 1</math> || <syntaxhighlight lang=mathematica>(m)/(n)*binomial(m - 1,n - 1) = (m - n + 1)/(n)*binomial(m,n - 1)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[m,n]*Binomial[m - 1,n - 1] == Divide[m - n + 1,n]*Binomial[m,n - 1]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 6]
-|-
-| [https://dlmf.nist.gov/26.3.E7 26.3.E7] || [[Item:Q7780|<math>\binom{m+1}{n+1} = \sum_{k=n}^{m}\binom{k}{n}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\binom{m+1}{n+1} = \sum_{k=n}^{m}\binom{k}{n}</syntaxhighlight> || <math>m \geq n, n \geq 0</math> || <syntaxhighlight lang=mathematica>binomial(m + 1,n + 1) = sum(binomial(k,n), k = n..m)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Binomial[m + 1,n + 1] == Sum[Binomial[k,n], {k, n, m}, GenerateConditions->None]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 6]
-|-
-| [https://dlmf.nist.gov/26.3.E8 26.3.E8] || [[Item:Q7781|<math>\binom{m}{n} = \sum_{k=0}^{n}\binom{m-n-1+k}{k}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\binom{m}{n} = \sum_{k=0}^{n}\binom{m-n-1+k}{k}</syntaxhighlight> || <math>m \geq n, n \geq 0</math> || <syntaxhighlight lang=mathematica>binomial(m,n) = sum(binomial(m - n - 1 + k,k), k = 0..n)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Binomial[m,n] == Sum[Binomial[m - n - 1 + k,k], {k, 0, n}, GenerateConditions->None]</syntaxhighlight> || Successful || Successful || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [3 / 6]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Indeterminate
-Test Values: {Rule[m, 1], Rule[n, 1]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Indeterminate
-Test Values: {Rule[m, 2], Rule[n, 2]}</syntaxhighlight><br>... skip entries to safe data</div></div>
-|-
-| [https://dlmf.nist.gov/26.3.E9 26.3.E9] || [[Item:Q7782|<math>\binom{n}{0} = \binom{n}{n}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\binom{n}{0} = \binom{n}{n}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>binomial(n,0) = binomial(n,n)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Binomial[n,0] == Binomial[n,n]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 3]
-|-
-| [https://dlmf.nist.gov/26.3.E9 26.3.E9] || [[Item:Q7782|<math>\binom{n}{n} = 1</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\binom{n}{n} = 1</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>binomial(n,n) = 1</syntaxhighlight> || <syntaxhighlight lang=mathematica>Binomial[n,n] == 1</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 3]
-|-
-| [https://dlmf.nist.gov/26.3.E10 26.3.E10] || [[Item:Q7783|<math>\binom{m}{n} = \sum_{k=0}^{n}(-1)^{n-k}\binom{m+1}{k}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\binom{m}{n} = \sum_{k=0}^{n}(-1)^{n-k}\binom{m+1}{k}</syntaxhighlight> || <math>m \geq n, n \geq 0</math> || <syntaxhighlight lang=mathematica>binomial(m,n) = sum((- 1)^(n - k)*binomial(m + 1,k), k = 0..n)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Binomial[m,n] == Sum[(- 1)^(n - k)*Binomial[m + 1,k], {k, 0, n}, GenerateConditions->None]</syntaxhighlight> || Successful || Failure || - || Successful [Tested: 6]
-|-
-| [https://dlmf.nist.gov/26.4.E1 26.4.E1] || [[Item:Q7786|<math>\multinomial{n_{1}+n_{2}}{n_{1},n_{2}} = \binom{n_{1}+n_{2}}{n_{1}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\multinomial{n_{1}+n_{2}}{n_{1},n_{2}} = \binom{n_{1}+n_{2}}{n_{1}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>multinomial(n[1]+ n[2], n[1], n[2]) = binomial(n[1]+ n[2],n[1])</syntaxhighlight> || <syntaxhighlight lang=mathematica>Multinomial[Subscript[n, 1]+ Subscript[n, 2]] == Binomial[Subscript[n, 1]+ Subscript[n, 2],Subscript[n, 1]]</syntaxhighlight> || Failure || Failure || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [100 / 100]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[-0.4855310647423219, -0.7913166384345096]
-Test Values: {Rule[Subscript[n, 1], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[n, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[0.5823425344168771, -0.5778520047366285]
-Test Values: {Rule[Subscript[n, 1], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[n, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
-|-
-| [https://dlmf.nist.gov/26.4.E1 26.4.E1] || [[Item:Q7786|<math>\binom{n_{1}+n_{2}}{n_{1}} = \binom{n_{1}+n_{2}}{n_{2}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\binom{n_{1}+n_{2}}{n_{1}} = \binom{n_{1}+n_{2}}{n_{2}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>binomial(n[1]+ n[2],n[1]) = binomial(n[1]+ n[2],n[2])</syntaxhighlight> || <syntaxhighlight lang=mathematica>Binomial[Subscript[n, 1]+ Subscript[n, 2],Subscript[n, 1]] == Binomial[Subscript[n, 1]+ Subscript[n, 2],Subscript[n, 2]]</syntaxhighlight> || Failure || Successful || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [6 / 100]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Indeterminate
-Test Values: {Rule[Subscript[n, 1], -1.5], Rule[Subscript[n, 2], -1.5]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Indeterminate
-Test Values: {Rule[Subscript[n, 1], -1.5], Rule[Subscript[n, 2], -0.5]}</syntaxhighlight><br>... skip entries to safe data</div></div>
-|-
-| [https://dlmf.nist.gov/26.5.E1 26.5.E1] || [[Item:Q7796|<math>\frac{1}{n+1}\binom{2n}{n} = \frac{1}{2n+1}\binom{2n+1}{n}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\frac{1}{n+1}\binom{2n}{n} = \frac{1}{2n+1}\binom{2n+1}{n}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(1)/(n + 1)*binomial(2*n,n) = (1)/(2*n + 1)*binomial(2*n + 1,n)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[1,n + 1]*Binomial[2*n,n] == Divide[1,2*n + 1]*Binomial[2*n + 1,n]</syntaxhighlight> || Successful || Successful || Skip - symbolical successful subtest || Successful [Tested: 3]
-|-
-| [https://dlmf.nist.gov/26.5.E1 26.5.E1] || [[Item:Q7796|<math>\frac{1}{2n+1}\binom{2n+1}{n} = \binom{2n}{n}-\binom{2n}{n-1}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\frac{1}{2n+1}\binom{2n+1}{n} = \binom{2n}{n}-\binom{2n}{n-1}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(1)/(2*n + 1)*binomial(2*n + 1,n) = binomial(2*n,n)-binomial(2*n,n - 1)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[1,2*n + 1]*Binomial[2*n + 1,n] == Binomial[2*n,n]-Binomial[2*n,n - 1]</syntaxhighlight> || Successful || Failure || Skip - symbolical successful subtest || Successful [Tested: 3]
-|-
-| [https://dlmf.nist.gov/26.5.E1 26.5.E1] || [[Item:Q7796|<math>\binom{2n}{n}-\binom{2n}{n-1} = \binom{2n-1}{n}-\binom{2n-1}{n+1}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\binom{2n}{n}-\binom{2n}{n-1} = \binom{2n-1}{n}-\binom{2n-1}{n+1}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>binomial(2*n,n)-binomial(2*n,n - 1) = binomial(2*n - 1,n)-binomial(2*n - 1,n + 1)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Binomial[2*n,n]-Binomial[2*n,n - 1] == Binomial[2*n - 1,n]-Binomial[2*n - 1,n + 1]</syntaxhighlight> || Failure || Failure || Successful [Tested: 3] || Successful [Tested: 3]
-|-
-| [https://dlmf.nist.gov/26.6.E5 26.6.E5] || [[Item:Q7807|<math>\sum_{m,n=0}^{\infty}D(m,n)x^{m}y^{n} = \frac{1}{1-x-y-xy}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sum_{m,n=0}^{\infty}D(m,n)x^{m}y^{n} = \frac{1}{1-x-y-xy}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>sum(sum((sum(binomial(n,k)*binomial(m + n - k,n), k = 0..n))*(x)^(m)* (y)^(n), n = 0..infinity), m = 0..infinity) = (1)/(1 - x - y - x*y)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sum[Sum[(Sum[Binomial[n,k]*Binomial[m + n - k,n], {k, 0, n}, GenerateConditions->None])*(x)^(m)* (y)^(n), {n, 0, Infinity}, GenerateConditions->None], {m, 0, Infinity}, GenerateConditions->None] == Divide[1,1 - x - y - x*y]</syntaxhighlight> || Skipped - no semantic math || Skipped - no semantic math || - || -
-|-
-| [https://dlmf.nist.gov/26.6.E6 26.6.E6] || [[Item:Q7808|<math>\sum_{n=0}^{\infty}D(n,n)x^{n} = \frac{1}{\sqrt{1-6x+x^{2}}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sum_{n=0}^{\infty}D(n,n)x^{n} = \frac{1}{\sqrt{1-6x+x^{2}}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>sum(D(n , n)* (x)^(n), n = 0..infinity) = (1)/(sqrt(1 - 6*x + (x)^(2)))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sum[D[n , n]* (x)^(n), {n, 0, Infinity}, GenerateConditions->None] == Divide[1,Sqrt[1 - 6*x + (x)^(2)]]</syntaxhighlight> || Skipped - no semantic math || Skipped - no semantic math || - || -
-|-
-| [https://dlmf.nist.gov/26.6.E7 26.6.E7] || [[Item:Q7809|<math>\sum_{n=0}^{\infty}M(n)x^{n} = \frac{1-x-\sqrt{1-2x-3x^{2}}}{2x^{2}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sum_{n=0}^{\infty}M(n)x^{n} = \frac{1-x-\sqrt{1-2x-3x^{2}}}{2x^{2}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>sum((sum(((- 1)^(k))/(n + 2 - k)*binomial(n,k)*binomial(2*n + 2 - 2*k,n + 1 - k), k = 0..n))*(x)^(n), n = 0..infinity) = (1 - x -sqrt(1 - 2*x - 3*(x)^(2)))/(2*(x)^(2))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sum[(Sum[Divide[(- 1)^(k),n + 2 - k]*Binomial[n,k]*Binomial[2*n + 2 - 2*k,n + 1 - k], {k, 0, n}, GenerateConditions->None])*(x)^(n), {n, 0, Infinity}, GenerateConditions->None] == Divide[1 - x -Sqrt[1 - 2*x - 3*(x)^(2)],2*(x)^(2)]</syntaxhighlight> || Skipped - no semantic math || Skipped - no semantic math || - || -
-|-
-| [https://dlmf.nist.gov/26.6.E8 26.6.E8] || [[Item:Q7810|<math>\sum_{n,k=1}^{\infty}N(n,k)x^{n}y^{k} = \frac{1-x-xy-\sqrt{(1-x-xy)^{2}-4x^{2}y}}{2x}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sum_{n,k=1}^{\infty}N(n,k)x^{n}y^{k} = \frac{1-x-xy-\sqrt{(1-x-xy)^{2}-4x^{2}y}}{2x}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>sum(sum(((1)/(n)*binomial(n,k)*binomial(n,k - 1))*(x)^(n)* (y)^(k), k = 1..infinity), n = 1..infinity) = (1 - x - x*y -sqrt((1 - x - x*y)^(2)- 4*(x)^(2)* y))/(2*x)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sum[Sum[(Divide[1,n]*Binomial[n,k]*Binomial[n,k - 1])*(x)^(n)* (y)^(k), {k, 1, Infinity}, GenerateConditions->None], {n, 1, Infinity}, GenerateConditions->None] == Divide[1 - x - x*y -Sqrt[(1 - x - x*y)^(2)- 4*(x)^(2)* y],2*x]</syntaxhighlight> || Skipped - no semantic math || Skipped - no semantic math || - || -
-|-
-| [https://dlmf.nist.gov/26.6.E9 26.6.E9] || [[Item:Q7811|<math>\sum_{n=0}^{\infty}r(n)x^{n} = \frac{1-x-\sqrt{1-6x+x^{2}}}{2x}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sum_{n=0}^{\infty}r(n)x^{n} = \frac{1-x-\sqrt{1-6x+x^{2}}}{2x}</syntaxhighlight> || <math>n \geq 1</math> || <syntaxhighlight lang=mathematica>sum((D(n , n)- D(n + 1 , n - 1))*(x)^(n), n = 0..infinity) = (1 - x -sqrt(1 - 6*x + (x)^(2)))/(2*x)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sum[(D[n , n]- D[n + 1 , n - 1])*(x)^(n), {n, 0, Infinity}, GenerateConditions->None] == Divide[1 - x -Sqrt[1 - 6*x + (x)^(2)],2*x]</syntaxhighlight> || Skipped - no semantic math || Skipped - no semantic math || - || -
-|-
-| [https://dlmf.nist.gov/26.6.E10 26.6.E10] || [[Item:Q7812|<math>D(m,n) = D(m,n-1)+D(m-1,n)+D(m-1,n-1)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>D(m,n) = D(m,n-1)+D(m-1,n)+D(m-1,n-1)</syntaxhighlight> || <math>m \geq 1, n \geq 1</math> || <syntaxhighlight lang=mathematica>(sum(binomial(n,k)*binomial(m + n - k,n), k = 0..n)) = D(m , n - 1)+ D(m - 1 , n)+ D(m - 1 , n - 1)</syntaxhighlight> || <syntaxhighlight lang=mathematica>(Sum[Binomial[n,k]*Binomial[m + n - k,n], {k, 0, n}, GenerateConditions->None]) == D[m , n - 1]+ D[m - 1 , n]+ D[m - 1 , n - 1]</syntaxhighlight> || Skipped - no semantic math || Skipped - no semantic math || - || -
-|-
-| [https://dlmf.nist.gov/26.6.E11 26.6.E11] || [[Item:Q7813|<math>M(n) = M(n-1)+\sum_{k=2}^{n}M(k-2)\,M(n-k)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>M(n) = M(n-1)+\sum_{k=2}^{n}M(k-2)\,M(n-k)</syntaxhighlight> || <math>n \geq 2</math> || <syntaxhighlight lang=mathematica>(sum(((- 1)^(k))/(n + 2 - k)*binomial(n,k)*binomial(2*n + 2 - 2*k,n + 1 - k), k = 0..n)) = M*(n - 1)+ sum(M*(k - 2)*M*(n - k), k = 2..n)</syntaxhighlight> || <syntaxhighlight lang=mathematica>(Sum[Divide[(- 1)^(k),n + 2 - k]*Binomial[n,k]*Binomial[2*n + 2 - 2*k,n + 1 - k], {k, 0, n}, GenerateConditions->None]) == M*(n - 1)+ Sum[M*(k - 2)*M*(n - k), {k, 2, n}, GenerateConditions->None]</syntaxhighlight> || Skipped - no semantic math || Skipped - no semantic math || - || -
-|-
-| [https://dlmf.nist.gov/26.7.E1 26.7.E1] || [[Item:Q7817|<math>\Bellnumber@{0} = 1</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Bellnumber@{0} = 1</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>BellB(0, 1) = 1</syntaxhighlight> || <syntaxhighlight lang=mathematica>BellB[0] == 1</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 1]
-|-
-| [https://dlmf.nist.gov/26.7.E2 26.7.E2] || [[Item:Q7818|<math>\Bellnumber@{n} = \sum_{k=0}^{n}\StirlingnumberS@{n}{k}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Bellnumber@{n} = \sum_{k=0}^{n}\StirlingnumberS@{n}{k}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>BellB(n, 1) = sum(Stirling2(n, k), k = 0..n)</syntaxhighlight> || <syntaxhighlight lang=mathematica>BellB[n] == Sum[StirlingS2[n, k], {k, 0, n}, GenerateConditions->None]</syntaxhighlight> || Failure || Successful || Successful [Tested: 3] || Successful [Tested: 3]
-|-
-| [https://dlmf.nist.gov/26.7.E3 26.7.E3] || [[Item:Q7819|<math>\Bellnumber@{n} = \sum_{k=1}^{m}\frac{k^{n}}{k!}\sum_{j=0}^{m-k}\frac{(-1)^{j}}{j!}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Bellnumber@{n} = \sum_{k=1}^{m}\frac{k^{n}}{k!}\sum_{j=0}^{m-k}\frac{(-1)^{j}}{j!}</syntaxhighlight> || <math>m \geq n</math> || <syntaxhighlight lang=mathematica>BellB(n, 1) = sum(((k)^(n))/(factorial(k))*sum(((- 1)^(j))/(factorial(j)), j = 0..m - k), k = 1..m)</syntaxhighlight> || <syntaxhighlight lang=mathematica>BellB[n] == Sum[Divide[(k)^(n),(k)!]*Sum[Divide[(- 1)^(j),(j)!], {j, 0, m - k}, GenerateConditions->None], {k, 1, m}, GenerateConditions->None]</syntaxhighlight> || Error || Failure || - || Successful [Tested: 6]
-|-
-| [https://dlmf.nist.gov/26.7.E4 26.7.E4] || [[Item:Q7820|<math>\Bellnumber@{n} = \expe^{-1}\sum_{k=1}^{\infty}\frac{k^{n}}{k!}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Bellnumber@{n} = \expe^{-1}\sum_{k=1}^{\infty}\frac{k^{n}}{k!}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>BellB(n, 1) = exp(- 1)*sum(((k)^(n))/(factorial(k)), k = 1..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>BellB[n] == Exp[- 1]*Sum[Divide[(k)^(n),(k)!], {k, 1, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Failure || Successful [Tested: 3] || Successful [Tested: 3]
-|-
-| [https://dlmf.nist.gov/26.7.E4 26.7.E4] || [[Item:Q7820|<math>\expe^{-1}\sum_{k=1}^{\infty}\frac{k^{n}}{k!} = 1+\floor{\expe^{-1}\sum_{k=1}^{2n}\frac{k^{n}}{k!}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\expe^{-1}\sum_{k=1}^{\infty}\frac{k^{n}}{k!} = 1+\floor{\expe^{-1}\sum_{k=1}^{2n}\frac{k^{n}}{k!}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>exp(- 1)*sum(((k)^(n))/(factorial(k)), k = 1..infinity) = 1 + floor(exp(- 1)*sum(((k)^(n))/(factorial(k)), k = 1..2*n))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Exp[- 1]*Sum[Divide[(k)^(n),(k)!], {k, 1, Infinity}, GenerateConditions->None] == 1 + Floor[Exp[- 1]*Sum[Divide[(k)^(n),(k)!], {k, 1, 2*n}, GenerateConditions->None]]</syntaxhighlight> || Failure || Failure || Successful [Tested: 3] || Successful [Tested: 3]
-|-
-| [https://dlmf.nist.gov/26.7.E5 26.7.E5] || [[Item:Q7821|<math>\sum_{n=0}^{\infty}\Bellnumber@{n}\frac{x^{n}}{n!} = \exp(\expe^{x}-1)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sum_{n=0}^{\infty}\Bellnumber@{n}\frac{x^{n}}{n!} = \exp(\expe^{x}-1)</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>sum(BellB(n, 1)*((x)^(n))/(factorial(n)), n = 0..infinity) = exp(exp(x)- 1)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sum[BellB[n]*Divide[(x)^(n),(n)!], {n, 0, Infinity}, GenerateConditions->None] == Exp[Exp[x]- 1]</syntaxhighlight> || Translation Error || Translation Error || - || -
-|-
-| [https://dlmf.nist.gov/26.7.E6 26.7.E6] || [[Item:Q7822|<math>\Bellnumber@{n+1} = \sum_{k=0}^{n}\binom{n}{k}\Bellnumber@{k}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Bellnumber@{n+1} = \sum_{k=0}^{n}\binom{n}{k}\Bellnumber@{k}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>BellB(n + 1, 1) = sum(binomial(n,k)*BellB(k, 1), k = 0..n)</syntaxhighlight> || <syntaxhighlight lang=mathematica>BellB[n + 1] == Sum[Binomial[n,k]*BellB[k], {k, 0, n}, GenerateConditions->None]</syntaxhighlight> || Failure || Failure || Successful [Tested: 3] || Successful [Tested: 3]
-|-
-| [https://dlmf.nist.gov/26.7#Ex1 26.7#Ex1] || [[Item:Q7823|<math>\Bellnumber@{n+1} = \sum_{k=0}^{n}\binom{n}{k}\Bellnumber@{n}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Bellnumber@{n+1} = \sum_{k=0}^{n}\binom{n}{k}\Bellnumber@{n}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>BellB(n + 1, 1) = sum(binomial(n,k)*BellB(n, 1), k = 0..n)</syntaxhighlight> || <syntaxhighlight lang=mathematica>BellB[n + 1] == Sum[Binomial[n,k]*BellB[n], {k, 0, n}, GenerateConditions->None]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [2 / 3]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: -3.
-Test Values: {n = 2}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -25.
-Test Values: {n = 3}</syntaxhighlight><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [2 / 3]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: -3.0
-Test Values: {Rule[n, 2]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -25.0
-Test Values: {Rule[n, 3]}</syntaxhighlight><br></div></div>
-|-
-| [https://dlmf.nist.gov/26.7.E8 26.7.E8] || [[Item:Q7825|<math>N\ln@@{N} = n</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>N\ln@@{N} = n</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>N*ln(N) = n</syntaxhighlight> || <syntaxhighlight lang=mathematica>N*Log[N] == n</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [30 / 30]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: -1.261799388+.4534498412*I
-Test Values: {N = 1/2*3^(1/2)+1/2*I, n = 1}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -2.261799388+.4534498412*I
-Test Values: {N = 1/2*3^(1/2)+1/2*I, n = 2}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [30 / 30]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[-1.2617993877991494, 0.4534498410585544]
-Test Values: {Rule[n, 1], Rule[N, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[-2.261799387799149, 0.4534498410585544]
-Test Values: {Rule[n, 2], Rule[N, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
-|-
-| [https://dlmf.nist.gov/26.8.E1 26.8.E1] || [[Item:Q7826|<math>\Stirlingnumbers@{n}{n} = 1</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Stirlingnumbers@{n}{n} = 1</syntaxhighlight> || <math>n \geq 0</math> || <syntaxhighlight lang=mathematica>Stirling1(n, n) = 1</syntaxhighlight> || <syntaxhighlight lang=mathematica>StirlingS1[n, n] == 1</syntaxhighlight> || Successful || Failure || - || Successful [Tested: 3]
-|-
-| [https://dlmf.nist.gov/26.8.E2 26.8.E2] || [[Item:Q7827|<math>\Stirlingnumbers@{1}{k} = \Kroneckerdelta{1}{k}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Stirlingnumbers@{1}{k} = \Kroneckerdelta{1}{k}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Stirling1(1, k) = KroneckerDelta[1, k]</syntaxhighlight> || <syntaxhighlight lang=mathematica>StirlingS1[1, k] == KroneckerDelta[1, k]</syntaxhighlight> || Failure || Failure || Successful [Tested: 3] || Successful [Tested: 3]
-|-
-| [https://dlmf.nist.gov/26.8.E4 26.8.E4] || [[Item:Q7829|<math>\StirlingnumberS@{n}{n} = 1</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\StirlingnumberS@{n}{n} = 1</syntaxhighlight> || <math>n \geq 0</math> || <syntaxhighlight lang=mathematica>Stirling2(n, n) = 1</syntaxhighlight> || <syntaxhighlight lang=mathematica>StirlingS2[n, n] == 1</syntaxhighlight> || Successful || Failure || - || Successful [Tested: 3]
-|-
-| [https://dlmf.nist.gov/26.8.E6 26.8.E6] || [[Item:Q7831|<math>\StirlingnumberS@{n}{k} = \frac{1}{k!}\sum_{j=0}^{k}(-1)^{k-j}\binom{k}{j}j^{n}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\StirlingnumberS@{n}{k} = \frac{1}{k!}\sum_{j=0}^{k}(-1)^{k-j}\binom{k}{j}j^{n}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Stirling2(n, k) = (1)/(factorial(k))*sum((- 1)^(k - j)*binomial(k,j)*(j)^(n), j = 0..k)</syntaxhighlight> || <syntaxhighlight lang=mathematica>StirlingS2[n, k] == Divide[1,(k)!]*Sum[(- 1)^(k - j)*Binomial[k,j]*(j)^(n), {j, 0, k}, GenerateConditions->None]</syntaxhighlight> || Aborted || Failure || Successful [Tested: 9] || Successful [Tested: 9]
-|-
-| [https://dlmf.nist.gov/26.8.E7 26.8.E7] || [[Item:Q7832|<math>\sum_{k=0}^{n}\Stirlingnumbers@{n}{k}x^{k} = (x-n+1)_{n}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sum_{k=0}^{n}\Stirlingnumbers@{n}{k}x^{k} = (x-n+1)_{n}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>sum(Stirling1(n, k)*(x)^(k), k = 0..n) = x - n + 1[n]</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sum[StirlingS1[n, k]*(x)^(k), {k, 0, n}, GenerateConditions->None] == Subscript[x - n + 1, n]</syntaxhighlight> || Failure || Failure || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [9 / 9]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[1.5, Times[-1.0, Subscript[1.5, 1]]]
-Test Values: {Rule[n, 1], Rule[x, 1.5]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Plus[0.75, Times[-1.0, Subscript[0.5, 2]]]
-Test Values: {Rule[n, 2], Rule[x, 1.5]}</syntaxhighlight><br>... skip entries to safe data</div></div>
-|-
-| [https://dlmf.nist.gov/26.8.E8 26.8.E8] || [[Item:Q7833|<math>\sum_{n=0}^{\infty}\Stirlingnumbers@{n}{k}\frac{x^{n}}{n!} = \frac{(\ln@{1+x})^{k}}{k!}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sum_{n=0}^{\infty}\Stirlingnumbers@{n}{k}\frac{x^{n}}{n!} = \frac{(\ln@{1+x})^{k}}{k!}</syntaxhighlight> || <math>|x| < 1</math> || <syntaxhighlight lang=mathematica>sum(Stirling1(n, k)*((x)^(n))/(factorial(n)), n = 0..infinity) = ((ln(1 + x))^(k))/(factorial(k))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sum[StirlingS1[n, k]*Divide[(x)^(n),(n)!], {n, 0, Infinity}, GenerateConditions->None] == Divide[(Log[1 + x])^(k),(k)!]</syntaxhighlight> || Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [2 / 3]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[-0.08220097694658271, NSum[Times[Power[0.5, n], Power[Factorial[n], -1], StirlingS1[n, 2]]
-Test Values: {n, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]], {Rule[k, 2], Rule[x, 0.5]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Plus[-0.011109876001414293, NSum[Times[Power[0.5, n], Power[Factorial[n], -1], StirlingS1[n, 3]]
-Test Values: {n, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]], {Rule[k, 3], Rule[x, 0.5]}</syntaxhighlight><br></div></div>
-|-
-| [https://dlmf.nist.gov/26.8.E9 26.8.E9] || [[Item:Q7834|<math>\sum_{n,k=0}^{\infty}\Stirlingnumbers@{n}{k}\frac{x^{n}}{n!}y^{k} = (1+x)^{y}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sum_{n,k=0}^{\infty}\Stirlingnumbers@{n}{k}\frac{x^{n}}{n!}y^{k} = (1+x)^{y}</syntaxhighlight> || <math>|x| < 1</math> || <syntaxhighlight lang=mathematica>sum(sum(Stirling1(n, k)*((x)^(n))/(factorial(n))*(y)^(k), k = 0..infinity), n = 0..infinity) = (1 + x)^(y)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sum[Sum[StirlingS1[n, k]*Divide[(x)^(n),(n)!]*(y)^(k), {k, 0, Infinity}, GenerateConditions->None], {n, 0, Infinity}, GenerateConditions->None] == (1 + x)^(y)</syntaxhighlight> || Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [6 / 6]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[-0.5443310539518174, NSum[Sum[Times[Power[-1.5, k], Power[0.5, n], Power[Factorial[n], -1], StirlingS1[n, k]]
-Test Values: {k, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]], {n, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]], {Rule[x, 0.5], Rule[y, -1.5]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Plus[-1.8371173070873836, NSum[Sum[Times[Power[0.5, n], Power[1.5, k], Power[Factorial[n], -1], StirlingS1[n, k]]
-Test Values: {k, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]], {n, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]], {Rule[x, 0.5], Rule[y, 1.5]}</syntaxhighlight><br>... skip entries to safe data</div></div>
-|-
-| [https://dlmf.nist.gov/26.8.E10 26.8.E10] || [[Item:Q7835|<math>\sum_{k=1}^{n}\StirlingnumberS@{n}{k}(x-k+1)_{k} = x^{n}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sum_{k=1}^{n}\StirlingnumberS@{n}{k}(x-k+1)_{k} = x^{n}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>sum(Stirling2(n, k)*x - k + 1[k], k = 1..n) = (x)^(n)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sum[StirlingS2[n, k]*Subscript[x - k + 1, k], {k, 1, n}, GenerateConditions->None] == (x)^(n)</syntaxhighlight> || Failure || Failure || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [9 / 9]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[-1.5, Subscript[1.5, 1]]
-Test Values: {Rule[n, 1], Rule[x, 1.5]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Plus[-2.25, Subscript[0.5, 2], Subscript[1.5, 1]]
-Test Values: {Rule[n, 2], Rule[x, 1.5]}</syntaxhighlight><br>... skip entries to safe data</div></div>
-|-
-| [https://dlmf.nist.gov/26.8.E12 26.8.E12] || [[Item:Q7837|<math>\sum_{n=0}^{\infty}\StirlingnumberS@{n}{k}\frac{x^{n}}{n!} = \frac{(\expe^{x}-1)^{k}}{k!}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sum_{n=0}^{\infty}\StirlingnumberS@{n}{k}\frac{x^{n}}{n!} = \frac{(\expe^{x}-1)^{k}}{k!}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>sum(Stirling2(n, k)*((x)^(n))/(factorial(n)), n = 0..infinity) = ((exp(x)- 1)^(k))/(factorial(k))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sum[StirlingS2[n, k]*Divide[(x)^(n),(n)!], {n, 0, Infinity}, GenerateConditions->None] == Divide[(Exp[x]- 1)^(k),(k)!]</syntaxhighlight> || Failure || Failure || Error || Successful [Tested: 9]
-|-
-| [https://dlmf.nist.gov/26.8.E13 26.8.E13] || [[Item:Q7838|<math>\sum_{n,k=0}^{\infty}\StirlingnumberS@{n}{k}\frac{x^{n}}{n!}y^{k} = \exp\left(y(\expe^{x}-1)\right)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sum_{n,k=0}^{\infty}\StirlingnumberS@{n}{k}\frac{x^{n}}{n!}y^{k} = \exp\left(y(\expe^{x}-1)\right)</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>sum(sum(Stirling2(n, k)*((x)^(n))/(factorial(n))*(y)^(k), k = 0..infinity), n = 0..infinity) = exp(y*(exp(x)- 1))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sum[Sum[StirlingS2[n, k]*Divide[(x)^(n),(n)!]*(y)^(k), {k, 0, Infinity}, GenerateConditions->None], {n, 0, Infinity}, GenerateConditions->None] == Exp[y*(Exp[x]- 1)]</syntaxhighlight> || Translation Error || Translation Error || - || -
-|-
-| [https://dlmf.nist.gov/26.8#Ex1 26.8#Ex1] || [[Item:Q7839|<math>\Stirlingnumbers@{n}{0} = 0</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Stirlingnumbers@{n}{0} = 0</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Stirling1(n, 0) = 0</syntaxhighlight> || <syntaxhighlight lang=mathematica>StirlingS1[n, 0] == 0</syntaxhighlight> || Failure || Failure || Successful [Tested: 3] || Successful [Tested: 3]
-|-
-| [https://dlmf.nist.gov/26.8#Ex2 26.8#Ex2] || [[Item:Q7840|<math>\Stirlingnumbers@{n}{1} = (-1)^{n-1}(n-1)!</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Stirlingnumbers@{n}{1} = (-1)^{n-1}(n-1)!</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Stirling1(n, 1) = (- 1)^(n - 1)*factorial(n - 1)</syntaxhighlight> || <syntaxhighlight lang=mathematica>StirlingS1[n, 1] == (- 1)^(n - 1)*(n - 1)!</syntaxhighlight> || Failure || Failure || Successful [Tested: 3] || Successful [Tested: 3]
-|-
-| [https://dlmf.nist.gov/26.8.E16 26.8.E16] || [[Item:Q7842|<math>-\Stirlingnumbers@{n}{n-1} = \StirlingnumberS@{n}{n-1}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>-\Stirlingnumbers@{n}{n-1} = \StirlingnumberS@{n}{n-1}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>- Stirling1(n, n - 1) = Stirling2(n, n - 1)</syntaxhighlight> || <syntaxhighlight lang=mathematica>- StirlingS1[n, n - 1] == StirlingS2[n, n - 1]</syntaxhighlight> || Successful || Failure || - || Successful [Tested: 3]
-|-
-| [https://dlmf.nist.gov/26.8.E16 26.8.E16] || [[Item:Q7842|<math>\StirlingnumberS@{n}{n-1} = \binom{n}{2}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\StirlingnumberS@{n}{n-1} = \binom{n}{2}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Stirling2(n, n - 1) = binomial(n,2)</syntaxhighlight> || <syntaxhighlight lang=mathematica>StirlingS2[n, n - 1] == Binomial[n,2]</syntaxhighlight> || Successful || Failure || - || Successful [Tested: 3]
-|-
-| [https://dlmf.nist.gov/26.8#Ex3 26.8#Ex3] || [[Item:Q7843|<math>\StirlingnumberS@{n}{0} = 0</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\StirlingnumberS@{n}{0} = 0</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Stirling2(n, 0) = 0</syntaxhighlight> || <syntaxhighlight lang=mathematica>StirlingS2[n, 0] == 0</syntaxhighlight> || Failure || Failure || Successful [Tested: 3] || Successful [Tested: 3]
-|-
-| [https://dlmf.nist.gov/26.8#Ex4 26.8#Ex4] || [[Item:Q7844|<math>\StirlingnumberS@{n}{1} = 1</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\StirlingnumberS@{n}{1} = 1</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Stirling2(n, 1) = 1</syntaxhighlight> || <syntaxhighlight lang=mathematica>StirlingS2[n, 1] == 1</syntaxhighlight> || Failure || Failure || Successful [Tested: 3] || Successful [Tested: 3]
-|-
-| [https://dlmf.nist.gov/26.8#Ex5 26.8#Ex5] || [[Item:Q7845|<math>\StirlingnumberS@{n}{2} = 2^{n-1}-1</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\StirlingnumberS@{n}{2} = 2^{n-1}-1</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Stirling2(n, 2) = (2)^(n - 1)- 1</syntaxhighlight> || <syntaxhighlight lang=mathematica>StirlingS2[n, 2] == (2)^(n - 1)- 1</syntaxhighlight> || Failure || Failure || Successful [Tested: 3] || Successful [Tested: 3]
-|-
-| [https://dlmf.nist.gov/26.8.E18 26.8.E18] || [[Item:Q7846|<math>\Stirlingnumbers@{n}{k} = \Stirlingnumbers@{n-1}{k-1}-(n-1)\Stirlingnumbers@{n-1}{k}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Stirlingnumbers@{n}{k} = \Stirlingnumbers@{n-1}{k-1}-(n-1)\Stirlingnumbers@{n-1}{k}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Stirling1(n, k) = Stirling1(n - 1, k - 1)-(n - 1)*Stirling1(n - 1, k)</syntaxhighlight> || <syntaxhighlight lang=mathematica>StirlingS1[n, k] == StirlingS1[n - 1, k - 1]-(n - 1)*StirlingS1[n - 1, k]</syntaxhighlight> || Failure || Failure || Successful [Tested: 9] || Successful [Tested: 9]
-|-
-| [https://dlmf.nist.gov/26.8.E19 26.8.E19] || [[Item:Q7847|<math>\binom{k}{h}\Stirlingnumbers@{n}{k} = \sum_{j=k-h}^{n-h}\binom{n}{j}\Stirlingnumbers@{n-j}{h}\Stirlingnumbers@{j}{k-h}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\binom{k}{h}\Stirlingnumbers@{n}{k} = \sum_{j=k-h}^{n-h}\binom{n}{j}\Stirlingnumbers@{n-j}{h}\Stirlingnumbers@{j}{k-h}</syntaxhighlight> || <math>n \geq k, k \geq h</math> || <syntaxhighlight lang=mathematica>binomial(k,h)*Stirling1(n, k) = sum(binomial(n,j)*Stirling1(n - j, h)*Stirling1(j, k - h), j = k - h..n - h)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Binomial[k,h]*StirlingS1[n, k] == Sum[Binomial[n,j]*StirlingS1[n - j, h]*StirlingS1[j, k - h], {j, k - h, n - h}, GenerateConditions->None]</syntaxhighlight> || Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [11 / 30]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: 0.16976527263135505
-Test Values: {Rule[h, -1.5], Rule[k, 1], Rule[n, 2]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -0.08488263631567752
-Test Values: {Rule[h, -1.5], Rule[k, 1], Rule[n, 3]}</syntaxhighlight><br>... skip entries to safe data</div></div>
-|-
-| [https://dlmf.nist.gov/26.8.E20 26.8.E20] || [[Item:Q7848|<math>\Stirlingnumbers@{n+1}{k+1} = n!\sum_{j=k}^{n}\frac{(-1)^{n-j}}{j!}\,\Stirlingnumbers@{j}{k}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Stirlingnumbers@{n+1}{k+1} = n!\sum_{j=k}^{n}\frac{(-1)^{n-j}}{j!}\,\Stirlingnumbers@{j}{k}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Stirling1(n + 1, k + 1) = factorial(n)*sum(((- 1)^(n - j))/(factorial(j))*Stirling1(j, k), j = k..n)</syntaxhighlight> || <syntaxhighlight lang=mathematica>StirlingS1[n + 1, k + 1] == (n)!*Sum[Divide[(- 1)^(n - j),(j)!]*StirlingS1[j, k], {j, k, n}, GenerateConditions->None]</syntaxhighlight> || Failure || Failure || Successful [Tested: 9] || Successful [Tested: 9]
-|-
-| [https://dlmf.nist.gov/26.8.E21 26.8.E21] || [[Item:Q7849|<math>\Stirlingnumbers@{n+k+1}{k} = -\sum_{j=0}^{k}(n+j)\Stirlingnumbers@{n+j}{j}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Stirlingnumbers@{n+k+1}{k} = -\sum_{j=0}^{k}(n+j)\Stirlingnumbers@{n+j}{j}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Stirling1(n + k + 1, k) = - sum((n + j)*Stirling1(n + j, j), j = 0..k)</syntaxhighlight> || <syntaxhighlight lang=mathematica>StirlingS1[n + k + 1, k] == - Sum[(n + j)*StirlingS1[n + j, j], {j, 0, k}, GenerateConditions->None]</syntaxhighlight> || Failure || Failure || Successful [Tested: 9] || Successful [Tested: 9]
-|-
-| [https://dlmf.nist.gov/26.8.E22 26.8.E22] || [[Item:Q7850|<math>\StirlingnumberS@{n}{k} = k\StirlingnumberS@{n-1}{k}+\StirlingnumberS@{n-1}{k-1}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\StirlingnumberS@{n}{k} = k\StirlingnumberS@{n-1}{k}+\StirlingnumberS@{n-1}{k-1}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Stirling2(n, k) = k*Stirling2(n - 1, k)+ Stirling2(n - 1, k - 1)</syntaxhighlight> || <syntaxhighlight lang=mathematica>StirlingS2[n, k] == k*StirlingS2[n - 1, k]+ StirlingS2[n - 1, k - 1]</syntaxhighlight> || Failure || Failure || Successful [Tested: 9] || Successful [Tested: 9]
-|-
-| [https://dlmf.nist.gov/26.8.E23 26.8.E23] || [[Item:Q7851|<math>\binom{k}{h}\StirlingnumberS@{n}{k} = \sum_{j=k-h}^{n-h}\binom{n}{j}\StirlingnumberS@{n-j}{h}\StirlingnumberS@{j}{k-h}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\binom{k}{h}\StirlingnumberS@{n}{k} = \sum_{j=k-h}^{n-h}\binom{n}{j}\StirlingnumberS@{n-j}{h}\StirlingnumberS@{j}{k-h}</syntaxhighlight> || <math>n \geq k, k \geq h</math> || <syntaxhighlight lang=mathematica>binomial(k,h)*Stirling2(n, k) = sum(binomial(n,j)*Stirling2(n - j, h)*Stirling2(j, k - h), j = k - h..n - h)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Binomial[k,h]*StirlingS2[n, k] == Sum[Binomial[n,j]*StirlingS2[n - j, h]*StirlingS2[j, k - h], {j, k - h, n - h}, GenerateConditions->None]</syntaxhighlight> || Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [22 / 30]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[-0.08488263631567752, Times[0.08488263631567751, StirlingS2[-1.5, -1.5], StirlingS2[2.5, 2.5]]]
-Test Values: {Rule[h, -1.5], Rule[k, 1], Rule[n, 1]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Plus[-0.08488263631567752, Times[-0.33953054526271004, StirlingS2[-0.5, -1.5], StirlingS2[2.5, 2.5]], Times[0.04850436360895858, StirlingS2[-1.5, -1.5], StirlingS2[3.5, 2.5]]]
-Test Values: {Rule[h, -1.5], Rule[k, 1], Rule[n, 2]}</syntaxhighlight><br>... skip entries to safe data</div></div>
-|-
-| [https://dlmf.nist.gov/26.8.E24 26.8.E24] || [[Item:Q7852|<math>\StirlingnumberS@{n}{k} = \sum_{j=k}^{n}\StirlingnumberS@{j-1}{k-1}k^{n-j}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\StirlingnumberS@{n}{k} = \sum_{j=k}^{n}\StirlingnumberS@{j-1}{k-1}k^{n-j}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Stirling2(n, k) = sum(Stirling2(j - 1, k - 1)*(k)^(n - j), j = k..n)</syntaxhighlight> || <syntaxhighlight lang=mathematica>StirlingS2[n, k] == Sum[StirlingS2[j - 1, k - 1]*(k)^(n - j), {j, k, n}, GenerateConditions->None]</syntaxhighlight> || Failure || Failure || Successful [Tested: 9] || Successful [Tested: 9]
-|-
-| [https://dlmf.nist.gov/26.8.E25 26.8.E25] || [[Item:Q7853|<math>\StirlingnumberS@{n+1}{k+1} = \sum_{j=k}^{n}\binom{n}{j}\StirlingnumberS@{j}{k}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\StirlingnumberS@{n+1}{k+1} = \sum_{j=k}^{n}\binom{n}{j}\StirlingnumberS@{j}{k}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Stirling2(n + 1, k + 1) = sum(binomial(n,j)*Stirling2(j, k), j = k..n)</syntaxhighlight> || <syntaxhighlight lang=mathematica>StirlingS2[n + 1, k + 1] == Sum[Binomial[n,j]*StirlingS2[j, k], {j, k, n}, GenerateConditions->None]</syntaxhighlight> || Failure || Failure || Successful [Tested: 9] || Successful [Tested: 9]
-|-
-| [https://dlmf.nist.gov/26.8.E26 26.8.E26] || [[Item:Q7854|<math>\StirlingnumberS@{n+k+1}{k} = \sum_{j=0}^{k}j\StirlingnumberS@{n+j}{j}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\StirlingnumberS@{n+k+1}{k} = \sum_{j=0}^{k}j\StirlingnumberS@{n+j}{j}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Stirling2(n + k + 1, k) = sum(j*Stirling2(n + j, j), j = 0..k)</syntaxhighlight> || <syntaxhighlight lang=mathematica>StirlingS2[n + k + 1, k] == Sum[j*StirlingS2[n + j, j], {j, 0, k}, GenerateConditions->None]</syntaxhighlight> || Failure || Successful || Successful [Tested: 9] || Successful [Tested: 9]
-|-
-| [https://dlmf.nist.gov/26.8.E27 26.8.E27] || [[Item:Q7855|<math>\Stirlingnumbers@{n}{n-k} = \sum_{j=0}^{k}(-1)^{j}\binom{n-1+j}{k+j}\,\binom{n+k}{k-j}\*\StirlingnumberS@{k+j}{j}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Stirlingnumbers@{n}{n-k} = \sum_{j=0}^{k}(-1)^{j}\binom{n-1+j}{k+j}\,\binom{n+k}{k-j}\*\StirlingnumberS@{k+j}{j}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Stirling1(n, n - k) = sum((- 1)^(j)*binomial(n - 1 + j,k + j)*binomial(n + k,k - j)* Stirling2(k + j, j), j = 0..k)</syntaxhighlight> || <syntaxhighlight lang=mathematica>StirlingS1[n, n - k] == Sum[(- 1)^(j)*Binomial[n - 1 + j,k + j]*Binomial[n + k,k - j]* StirlingS2[k + j, j], {j, 0, k}, GenerateConditions->None]</syntaxhighlight> || Failure || Failure || Successful [Tested: 9] || <div class="toccolours mw-collapsible mw-collapsed">Failed [3 / 9]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: StirlingS1[1.0, -1.0]
-Test Values: {Rule[k, 2], Rule[n, 1]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: StirlingS1[1.0, -2.0]
-Test Values: {Rule[k, 3], Rule[n, 1]}</syntaxhighlight><br>... skip entries to safe data</div></div>
-|-
-| [https://dlmf.nist.gov/26.8.E28 26.8.E28] || [[Item:Q7856|<math>\sum_{k=1}^{n}\Stirlingnumbers@{n}{k} = 0</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sum_{k=1}^{n}\Stirlingnumbers@{n}{k} = 0</syntaxhighlight> || <math>n > 1</math> || <syntaxhighlight lang=mathematica>sum(Stirling1(n, k), k = 1..n) = 0</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sum[StirlingS1[n, k], {k, 1, n}, GenerateConditions->None] == 0</syntaxhighlight> || Failure || Failure || Successful [Tested: 2] || Successful [Tested: 2]
-|-
-| [https://dlmf.nist.gov/26.8.E29 26.8.E29] || [[Item:Q7857|<math>\sum_{k=1}^{n}(-1)^{n-k}\Stirlingnumbers@{n}{k} = n!</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sum_{k=1}^{n}(-1)^{n-k}\Stirlingnumbers@{n}{k} = n!</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>sum((- 1)^(n - k)* Stirling1(n, k), k = 1..n) = factorial(n)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sum[(- 1)^(n - k)* StirlingS1[n, k], {k, 1, n}, GenerateConditions->None] == (n)!</syntaxhighlight> || Failure || Failure || Successful [Tested: 3] || Successful [Tested: 3]
-|-
-| [https://dlmf.nist.gov/26.8.E30 26.8.E30] || [[Item:Q7858|<math>\sum_{j=k}^{n}\Stirlingnumbers@{n+1}{j+1}\,n^{j-k} = \Stirlingnumbers@{n}{k}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sum_{j=k}^{n}\Stirlingnumbers@{n+1}{j+1}\,n^{j-k} = \Stirlingnumbers@{n}{k}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>sum(Stirling1(n + 1, j + 1)*(n)^(j - k), j = k..n) = Stirling1(n, k)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sum[StirlingS1[n + 1, j + 1]*(n)^(j - k), {j, k, n}, GenerateConditions->None] == StirlingS1[n, k]</syntaxhighlight> || Failure || Successful || Successful [Tested: 9] || Successful [Tested: 9]
-|-
-| [https://dlmf.nist.gov/26.8.E33 26.8.E33] || [[Item:Q7861|<math>\StirlingnumberS@{n}{n-k} = \sum_{j=0}^{k}(-1)^{j}\binom{n-1+j}{k+j}\binom{n+k}{k-j}\*\Stirlingnumbers@{k+j}{j}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\StirlingnumberS@{n}{n-k} = \sum_{j=0}^{k}(-1)^{j}\binom{n-1+j}{k+j}\binom{n+k}{k-j}\*\Stirlingnumbers@{k+j}{j}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Stirling2(n, n - k) = sum((- 1)^(j)*binomial(n - 1 + j,k + j)*binomial(n + k,k - j)* Stirling1(k + j, j), j = 0..k)</syntaxhighlight> || <syntaxhighlight lang=mathematica>StirlingS2[n, n - k] == Sum[(- 1)^(j)*Binomial[n - 1 + j,k + j]*Binomial[n + k,k - j]* StirlingS1[k + j, j], {j, 0, k}, GenerateConditions->None]</syntaxhighlight> || Failure || Failure || Successful [Tested: 9] || <div class="toccolours mw-collapsible mw-collapsed">Failed [3 / 9]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: StirlingS2[1.0, -1.0]
-Test Values: {Rule[k, 2], Rule[n, 1]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: StirlingS2[1.0, -2.0]
-Test Values: {Rule[k, 3], Rule[n, 1]}</syntaxhighlight><br>... skip entries to safe data</div></div>
-|-
-| [https://dlmf.nist.gov/26.8.E34 26.8.E34] || [[Item:Q7862|<math>\sum_{j=0}^{n}j^{k}x^{j} = \sum_{j=0}^{k}\StirlingnumberS@{k}{j}x^{j}\deriv[j]{}{x}\left(\frac{1-x^{n+1}}{1-x}\right)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sum_{j=0}^{n}j^{k}x^{j} = \sum_{j=0}^{k}\StirlingnumberS@{k}{j}x^{j}\deriv[j]{}{x}\left(\frac{1-x^{n+1}}{1-x}\right)</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>sum((j)^(k)* (x)^(j), j = 0..n) = sum(Stirling2(k, j)*(x)^(j)* diff((1 - (x)^(n + 1))/(1 - x), [x$(j)]), j = 0..k)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sum[(j)^(k)* (x)^(j), {j, 0, n}, GenerateConditions->None] == Sum[StirlingS2[k, j]*(x)^(j)* D[Divide[1 - (x)^(n + 1),1 - x], {x, j}], {j, 0, k}, GenerateConditions->None]</syntaxhighlight> || Aborted || Failure || Skipped - Because timed out || Skipped - Because timed out
-|-
-| [https://dlmf.nist.gov/26.8.E35 26.8.E35] || [[Item:Q7863|<math>\sum_{j=0}^{n}j^{k} = \sum_{j=0}^{k}j!\StirlingnumberS@{k}{j}\binom{n+1}{j+1}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sum_{j=0}^{n}j^{k} = \sum_{j=0}^{k}j!\StirlingnumberS@{k}{j}\binom{n+1}{j+1}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>sum((j)^(k), j = 0..n) = sum(factorial(j)*Stirling2(k, j)*binomial(n + 1,j + 1), j = 0..k)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sum[(j)^(k), {j, 0, n}, GenerateConditions->None] == Sum[(j)!*StirlingS2[k, j]*Binomial[n + 1,j + 1], {j, 0, k}, GenerateConditions->None]</syntaxhighlight> || Failure || Failure || Successful [Tested: 9] || Successful [Tested: 9]
-|-
-| [https://dlmf.nist.gov/26.8.E36 26.8.E36] || [[Item:Q7864|<math>\sum_{k=0}^{n}(-1)^{n-k}k!\StirlingnumberS@{n}{k} = 1</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sum_{k=0}^{n}(-1)^{n-k}k!\StirlingnumberS@{n}{k} = 1</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>sum((- 1)^(n - k)* factorial(k)*Stirling2(n, k), k = 0..n) = 1</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sum[(- 1)^(n - k)* (k)!*StirlingS2[n, k], {k, 0, n}, GenerateConditions->None] == 1</syntaxhighlight> || Failure || Failure || Successful [Tested: 3] || Successful [Tested: 3]
-|-
-| [https://dlmf.nist.gov/26.8.E38 26.8.E38] || [[Item:Q7866|<math>A^{-1} = B</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>A^{-1} = B</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(A)^(- 1) = B</syntaxhighlight> || <syntaxhighlight lang=mathematica>(A)^(- 1) == B</syntaxhighlight> || Skipped - no semantic math || Skipped - no semantic math || - || -
-|-
-| [https://dlmf.nist.gov/26.8.E39 26.8.E39] || [[Item:Q7867|<math>\sum_{j=k}^{n}\Stirlingnumbers@{j}{k}\StirlingnumberS@{n}{j} = \sum_{j=k}^{n}\Stirlingnumbers@{n}{j}\StirlingnumberS@{j}{k}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sum_{j=k}^{n}\Stirlingnumbers@{j}{k}\StirlingnumberS@{n}{j} = \sum_{j=k}^{n}\Stirlingnumbers@{n}{j}\StirlingnumberS@{j}{k}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>sum(Stirling1(j, k)*Stirling2(n, j), j = k..n) = sum(Stirling1(n, j)*Stirling2(j, k), j = k..n)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sum[StirlingS1[j, k]*StirlingS2[n, j], {j, k, n}, GenerateConditions->None] == Sum[StirlingS1[n, j]*StirlingS2[j, k], {j, k, n}, GenerateConditions->None]</syntaxhighlight> || Failure || Failure || Successful [Tested: 9] || Successful [Tested: 9]
-|-
-| [https://dlmf.nist.gov/26.8.E39 26.8.E39] || [[Item:Q7867|<math>\sum_{j=k}^{n}\Stirlingnumbers@{n}{j}\StirlingnumberS@{j}{k} = \Kroneckerdelta{n}{k}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sum_{j=k}^{n}\Stirlingnumbers@{n}{j}\StirlingnumberS@{j}{k} = \Kroneckerdelta{n}{k}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>sum(Stirling1(n, j)*Stirling2(j, k), j = k..n) = KroneckerDelta[n, k]</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sum[StirlingS1[n, j]*StirlingS2[j, k], {j, k, n}, GenerateConditions->None] == KroneckerDelta[n, k]</syntaxhighlight> || Failure || Failure || Successful [Tested: 9] || Successful [Tested: 9]
-|-
-| [https://dlmf.nist.gov/26.9.E4 26.9.E4] || [[Item:Q7877|<math>\qbinom{m}{n}{q} = \prod_{j=1}^{n}\frac{1-q^{m-n+j}}{1-q^{j}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\qbinom{m}{n}{q} = \prod_{j=1}^{n}\frac{1-q^{m-n+j}}{1-q^{j}}</syntaxhighlight> || <math>n \geq 0</math> || <syntaxhighlight lang=mathematica>QBinomial(m, n, q) = product((1 - (q)^(m - n + j))/(1 - (q)^(j)), j = 1..n)</syntaxhighlight> || <syntaxhighlight lang=mathematica>QBinomial[m,n,q] == Product[Divide[1 - (q)^(m - n + j),1 - (q)^(j)], {j, 1, n}, GenerateConditions->None]</syntaxhighlight> || Failure || Failure || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [32 / 90]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: DirectedInfinity[]
-Test Values: {Rule[m, 1], Rule[n, 1], Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: DirectedInfinity[]
-Test Values: {Rule[m, 2], Rule[n, 2], Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
-|-
-| [https://dlmf.nist.gov/26.9.E5 26.9.E5] || [[Item:Q7878|<math>\prod_{j=1}^{k}\frac{1}{1-q^{j}} = 1+\sum_{m=1}^{\infty}\qbinom{k+m-1}{m}{q}q^{m}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\prod_{j=1}^{k}\frac{1}{1-q^{j}} = 1+\sum_{m=1}^{\infty}\qbinom{k+m-1}{m}{q}q^{m}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>product((1)/(1 - (q)^(j)), j = 1..k) = 1 + sum(QBinomial(k + m - 1, m, q)*(q)^(m), m = 1..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Product[Divide[1,1 - (q)^(j)], {j, 1, k}, GenerateConditions->None] == 1 + Sum[QBinomial[k + m - 1,m,q]*(q)^(m), {m, 1, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Aborted || Error || Skipped - Because timed out
-|-
-| [https://dlmf.nist.gov/26.9.E7 26.9.E7] || [[Item:Q7880|<math>1+\sum_{k=1}^{\infty}\qbinom{m+k}{k}{q}x^{k} = \prod_{j=0}^{m}\frac{1}{1-x\,q^{j}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>1+\sum_{k=1}^{\infty}\qbinom{m+k}{k}{q}x^{k} = \prod_{j=0}^{m}\frac{1}{1-x\,q^{j}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>1 + sum(QBinomial(m + k, k, q)*(x)^(k), k = 1..infinity) = product((1)/(1 - x*(q)^(j)), j = 0..m)</syntaxhighlight> || <syntaxhighlight lang=mathematica>1 + Sum[QBinomial[m + k,k,q]*(x)^(k), {k, 1, Infinity}, GenerateConditions->None] == Product[Divide[1,1 - x*(q)^(j)], {j, 0, m}, GenerateConditions->None]</syntaxhighlight> || Failure || Aborted || Error || Skipped - Because timed out
-|-
-| [https://dlmf.nist.gov/26.10.E2 26.10.E2] || [[Item:Q7885|<math>\prod_{j=1}^{\infty}(1+q^{j}) = \prod_{j=1}^{\infty}\frac{1}{1-q^{2j-1}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\prod_{j=1}^{\infty}(1+q^{j}) = \prod_{j=1}^{\infty}\frac{1}{1-q^{2j-1}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>product(1 + (q)^(j), j = 1..infinity) = product((1)/(1 - (q)^(2*j - 1)), j = 1..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Product[1 + (q)^(j), {j, 1, Infinity}, GenerateConditions->None] == Product[Divide[1,1 - (q)^(2*j - 1)], {j, 1, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Failure || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [1 / 10]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: DirectedInfinity[]
-Test Values: {Rule[q, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br></div></div>
-|-
-| [https://dlmf.nist.gov/26.10.E3 26.10.E3] || [[Item:Q7886|<math>\sum_{m=0}^{k}\qbinom{k}{m}{q}q^{m(m+1)/2}x^{m} = \prod_{j=1}^{k}(1+x\,q^{j})</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sum_{m=0}^{k}\qbinom{k}{m}{q}q^{m(m+1)/2}x^{m} = \prod_{j=1}^{k}(1+x\,q^{j})</syntaxhighlight> || <math>|x| < 1</math> || <syntaxhighlight lang=mathematica>sum(QBinomial(k, m, q)*(q)^(m*(m + 1)/2)* (x)^(m), m = 0..k) = product(1 + x*(q)^(j), j = 1..k)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sum[QBinomial[k,m,q]*(q)^(m*(m + 1)/2)* (x)^(m), {m, 0, k}, GenerateConditions->None] == Product[1 + x*(q)^(j), {j, 1, k}, GenerateConditions->None]</syntaxhighlight> || Failure || Failure || Error || Successful [Tested: 30]
-|-
-| [https://dlmf.nist.gov/26.11.E2 26.11.E2] || [[Item:Q7905|<math>\ncompositions[m]@{0} = \Kroneckerdelta{0}{m}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\ncompositions[m]@{0} = \Kroneckerdelta{0}{m}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>numbcomp(0, m) = KroneckerDelta[0, m]</syntaxhighlight> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || Error || Missing Macro Error || - || -
-|-
-| [https://dlmf.nist.gov/26.11.E3 26.11.E3] || [[Item:Q7906|<math>\ncompositions[m]@{n} = \binom{n-1}{m-1}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\ncompositions[m]@{n} = \binom{n-1}{m-1}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>numbcomp(n, m) = binomial(n - 1,m - 1)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || Error || Missing Macro Error || - || -
-|-
-| [https://dlmf.nist.gov/26.11.E4 26.11.E4] || [[Item:Q7907|<math>\sum_{n=0}^{\infty}\ncompositions[m]@{n}q^{n} = \frac{q^{m}}{(1-q)^{m}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sum_{n=0}^{\infty}\ncompositions[m]@{n}q^{n} = \frac{q^{m}}{(1-q)^{m}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>sum(numbcomp(n, m)*(q)^(n), n = 0..infinity) = ((q)^(m))/((1 - q)^(m))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || Error || Missing Macro Error || - || -
-|-
-| [https://dlmf.nist.gov/26.11#Ex1 26.11#Ex1] || [[Item:Q7908|<math>F_{0} = 0</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>F_{0} = 0</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>F[0] = 0</syntaxhighlight> || <syntaxhighlight lang=mathematica>Subscript[F, 0] == 0</syntaxhighlight> || Skipped - no semantic math || Skipped - no semantic math || - || -
-|-
-| [https://dlmf.nist.gov/26.11#Ex2 26.11#Ex2] || [[Item:Q7909|<math>F_{1} = 1</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>F_{1} = 1</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>F[1] = 1</syntaxhighlight> || <syntaxhighlight lang=mathematica>Subscript[F, 1] == 1</syntaxhighlight> || Skipped - no semantic math || Skipped - no semantic math || - || -
-|-
-| [https://dlmf.nist.gov/26.11#Ex3 26.11#Ex3] || [[Item:Q7910|<math>F_{n} = F_{n-1}+F_{n-2}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>F_{n} = F_{n-1}+F_{n-2}</syntaxhighlight> || <math>n \geq 2</math> || <syntaxhighlight lang=mathematica>F[n] = F[n - 1]+ F[n - 2]</syntaxhighlight> || <syntaxhighlight lang=mathematica>Subscript[F, n] == Subscript[F, n - 1]+ Subscript[F, n - 2]</syntaxhighlight> || Skipped - no semantic math || Skipped - no semantic math || - || -
-|-
-| [https://dlmf.nist.gov/26.11.E7 26.11.E7] || [[Item:Q7912|<math>F_{n} = \frac{(1+\sqrt{5})^{n}-(1-\sqrt{5})^{n}}{2^{n}\,\sqrt{5}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>F_{n} = \frac{(1+\sqrt{5})^{n}-(1-\sqrt{5})^{n}}{2^{n}\,\sqrt{5}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>F[n] = ((1 +sqrt(5))^(n)-(1 -sqrt(5))^(n))/((2)^(n)*sqrt(5))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Subscript[F, n] == Divide[(1 +Sqrt[5])^(n)-(1 -Sqrt[5])^(n),(2)^(n)*Sqrt[5]]</syntaxhighlight> || Skipped - no semantic math || Skipped - no semantic math || - || -
-|-
-| [https://dlmf.nist.gov/26.12.E23 26.12.E23] || [[Item:Q7940|<math>\prod_{h=1}^{r}\frac{1-q^{3h-1}}{1-q^{3h-2}}\prod_{1\leq h<j\leq r}\frac{1-q^{3(h+2j-1)}}{1-q^{3(h+j-1)}} = \prod_{h=1}^{r}\left(\frac{1-q^{3h-1}}{1-q^{3h-2}}\prod_{j=h}^{r}\frac{1-q^{3(r+h+j-1)}}{1-q^{3(2h+j-1)}}\right)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\prod_{h=1}^{r}\frac{1-q^{3h-1}}{1-q^{3h-2}}\prod_{1\leq h<j\leq r}\frac{1-q^{3(h+2j-1)}}{1-q^{3(h+j-1)}} = \prod_{h=1}^{r}\left(\frac{1-q^{3h-1}}{1-q^{3h-2}}\prod_{j=h}^{r}\frac{1-q^{3(r+h+j-1)}}{1-q^{3(2h+j-1)}}\right)</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>product((1 - (q)^(3*h - 1))/(1 - (q)^(3*h - 2)), h = 1..r)*product(product((1 - (q)^(3*(h + 2*j - 1)))/(1 - (q)^(3*(h + j - 1))), j = h + 1..r), h = 1..j - 1) = product((1 - (q)^(3*h - 1))/(1 - (q)^(3*h - 2))*product((1 - (q)^(3*(r + h + j - 1)))/(1 - (q)^(3*(2*h + j - 1))), j = h..r), h = 1..r)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Product[Divide[1 - (q)^(3*h - 1),1 - (q)^(3*h - 2)], {h, 1, r}, GenerateConditions->None]*Product[Product[Divide[1 - (q)^(3*(h + 2*j - 1)),1 - (q)^(3*(h + j - 1))], {j, h + 1, r}, GenerateConditions->None], {h, 1, j - 1}, GenerateConditions->None] == Product[Divide[1 - (q)^(3*h - 1),1 - (q)^(3*h - 2)]*Product[Divide[1 - (q)^(3*(r + h + j - 1)),1 - (q)^(3*(2*h + j - 1))], {j, h, r}, GenerateConditions->None], {h, 1, r}, GenerateConditions->None]</syntaxhighlight> || Failure || Aborted || Skipped - Because timed out || Skipped - Because timed out
-|-
-| [https://dlmf.nist.gov/26.12#Ex7 26.12#Ex7] || [[Item:Q7944|<math>\Riemannzeta@{3} = 1.20205\;69032</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Riemannzeta@{3} = 1.20205\;69032</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Zeta(3) = 1.2020569032</syntaxhighlight> || <syntaxhighlight lang=mathematica>Zeta[3] == 1.2020569032</syntaxhighlight> || Successful || Failure || - || Successful [Tested: 1]
-|-
-| [https://dlmf.nist.gov/26.12#Ex8 26.12#Ex8] || [[Item:Q7945|<math>\Riemannzeta'@{-1} = -0.16542\;11437</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Riemannzeta'@{-1} = -0.16542\;11437</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>subs( temp=- 1, diff( Zeta(temp), temp$(1) ) ) = - 0.1654211437</syntaxhighlight> || <syntaxhighlight lang=mathematica>(D[Zeta[temp], {temp, 1}]/.temp-> - 1) == - 0.1654211437</syntaxhighlight> || Successful || Failure || - || Successful [Tested: 1]
-|-
-| [https://dlmf.nist.gov/26.13.E4 26.13.E4] || [[Item:Q7949|<math>d(n) = n!\sum_{j=0}^{n}(-1)^{j}\frac{1}{j!}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>d(n) = n!\sum_{j=0}^{n}(-1)^{j}\frac{1}{j!}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>d(n) = factorial(n)*sum((- 1)^(j)*(1)/(factorial(j)), j = 0..n)</syntaxhighlight> || <syntaxhighlight lang=mathematica>d[n] == (n)!*Sum[(- 1)^(j)*Divide[1,(j)!], {j, 0, n}, GenerateConditions->None]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [29 / 30]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: .8660254040+.5000000000*I
-Test Values: {d = 1/2*3^(1/2)+1/2*I, n = 1}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: .7320508081+1.*I
-Test Values: {d = 1/2*3^(1/2)+1/2*I, n = 2}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [29 / 30]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[0.8660254037844387, 0.49999999999999994]
-Test Values: {Rule[d, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[n, 1]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[0.7320508075688774, 0.9999999999999999]
-Test Values: {Rule[d, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[n, 2]}</syntaxhighlight><br>... skip entries to safe data</div></div>
-|-
-| [https://dlmf.nist.gov/26.13.E4 26.13.E4] || [[Item:Q7949|<math>n!\sum_{j=0}^{n}(-1)^{j}\frac{1}{j!} = \floor{\frac{n!+\expe-2}{\expe}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>n!\sum_{j=0}^{n}(-1)^{j}\frac{1}{j!} = \floor{\frac{n!+\expe-2}{\expe}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>factorial(n)*sum((- 1)^(j)*(1)/(factorial(j)), j = 0..n) = floor((factorial(n)+ exp(1)- 2)/(exp(1)))</syntaxhighlight> || <syntaxhighlight lang=mathematica>(n)!*Sum[(- 1)^(j)*Divide[1,(j)!], {j, 0, n}, GenerateConditions->None] == Floor[Divide[(n)!+ E - 2,E]]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [1 / 3]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: .9999999999
-Test Values: {n = 2}</syntaxhighlight><br></div></div> || Successful [Tested: 3]
-|-
-| [https://dlmf.nist.gov/26.14.E4 26.14.E4] || [[Item:Q7955|<math>\sum_{n,k=0}^{\infty}\Euleriannumber{n}{k}x^{k}\,\frac{t^{n}}{n!} = \frac{1-x}{\exp((x-1)t)-x}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sum_{n,k=0}^{\infty}\Euleriannumber{n}{k}x^{k}\,\frac{t^{n}}{n!} = \frac{1-x}{\exp((x-1)t)-x}</syntaxhighlight> || <math>|x| < 1, |t| < 1</math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sum[Sum[Sum[(-1)^m Binomial[n+1,m] (k-m+1)^(n),{m,0,k+1}]*(x)^(k)*Divide[(t)^(n),(n)!], {k, 0, Infinity}, GenerateConditions->None], {n, 0, Infinity}, GenerateConditions->None] == Divide[1 - x,Exp[(x - 1)*t]- x]</syntaxhighlight> || Missing Macro Error || Translation Error || - || -
-|-
-| [https://dlmf.nist.gov/26.14.E5 26.14.E5] || [[Item:Q7956|<math>\sum_{k=0}^{n-1}\Euleriannumber{n}{k}\binom{x+k}{n} = x^{n}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sum_{k=0}^{n-1}\Euleriannumber{n}{k}\binom{x+k}{n} = x^{n}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sum[Sum[(-1)^m Binomial[n+1,m] (k-m+1)^(n),{m,0,k+1}]*Binomial[x + k,n], {k, 0, n - 1}, GenerateConditions->None] == (x)^(n)</syntaxhighlight> || Missing Macro Error || Failure || - || Successful [Tested: 9]
-|-
-| [https://dlmf.nist.gov/26.14.E6 26.14.E6] || [[Item:Q7957|<math>\Euleriannumber{n}{k} = \sum_{j=0}^{k}(-1)^{j}\binom{n+1}{j}(k+1-j)^{n}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Euleriannumber{n}{k} = \sum_{j=0}^{k}(-1)^{j}\binom{n+1}{j}(k+1-j)^{n}</syntaxhighlight> || <math>n \geq 1</math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sum[(-1)^m Binomial[n+1,m] (k-m+1)^(n),{m,0,k+1}] == Sum[(- 1)^(j)*Binomial[n + 1,j]*(k + 1 - j)^(n), {j, 0, k}, GenerateConditions->None]</syntaxhighlight> || Missing Macro Error || Failure || - || Successful [Tested: 9]
-|-
-| [https://dlmf.nist.gov/26.14.E7 26.14.E7] || [[Item:Q7958|<math>\Euleriannumber{n}{k} = \sum_{j=0}^{n-k}(-1)^{n-k-j}j!\binom{n-j}{k}\StirlingnumberS@{n}{j}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Euleriannumber{n}{k} = \sum_{j=0}^{n-k}(-1)^{n-k-j}j!\binom{n-j}{k}\StirlingnumberS@{n}{j}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sum[(-1)^m Binomial[n+1,m] (k-m+1)^(n),{m,0,k+1}] == Sum[(- 1)^(n - k - j)* (j)!*Binomial[n - j,k]*StirlingS2[n, j], {j, 0, n - k}, GenerateConditions->None]</syntaxhighlight> || Missing Macro Error || Failure || - || Successful [Tested: 9]
-|-
-| [https://dlmf.nist.gov/26.14.E8 26.14.E8] || [[Item:Q7959|<math>\Euleriannumber{n}{k} = (k+1)\Euleriannumber{n-1}{k}+(n-k)\Euleriannumber{n-1}{k-1}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Euleriannumber{n}{k} = (k+1)\Euleriannumber{n-1}{k}+(n-k)\Euleriannumber{n-1}{k-1}</syntaxhighlight> || <math>n \geq 2</math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sum[(-1)^m Binomial[n+1,m] (k-m+1)^(n),{m,0,k+1}] == (k + 1)*Sum[(-1)^m Binomial[n - 1+1,m] (k-m+1)^(n - 1),{m,0,k+1}]+(n - k)*Sum[(-1)^m Binomial[n - 1+1,m] (k - 1-m+1)^(n - 1),{m,0,k - 1+1}]</syntaxhighlight> || Missing Macro Error || Failure || - || Successful [Tested: 6]
-|-
-| [https://dlmf.nist.gov/26.14.E9 26.14.E9] || [[Item:Q7960|<math>\Euleriannumber{n}{k} = \Euleriannumber{n}{n-1-k}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Euleriannumber{n}{k} = \Euleriannumber{n}{n-1-k}</syntaxhighlight> || <math>n \geq 1</math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sum[(-1)^m Binomial[n+1,m] (k-m+1)^(n),{m,0,k+1}] == Sum[(-1)^m Binomial[n+1,m] (n - 1 - k-m+1)^(n),{m,0,n - 1 - k+1}]</syntaxhighlight> || Missing Macro Error || Failure || - || Successful [Tested: 9]
-|-
-| [https://dlmf.nist.gov/26.14.E10 26.14.E10] || [[Item:Q7961|<math>\sum_{k=0}^{n-1}\Euleriannumber{n}{k} = n!</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sum_{k=0}^{n-1}\Euleriannumber{n}{k} = n!</syntaxhighlight> || <math>n \geq 1</math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sum[Sum[(-1)^m Binomial[n+1,m] (k-m+1)^(n),{m,0,k+1}], {k, 0, n - 1}, GenerateConditions->None] == (n)!</syntaxhighlight> || Missing Macro Error || Failure || - || Successful [Tested: 3]
-|-
-| [https://dlmf.nist.gov/26.14.E11 26.14.E11] || [[Item:Q7962|<math>\BernoullinumberB{m} = \frac{m}{2^{m}(2^{m}-1)}\sum_{k=0}^{m-2}(-1)^{k}\Euleriannumber{m-1}{k}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\BernoullinumberB{m} = \frac{m}{2^{m}(2^{m}-1)}\sum_{k=0}^{m-2}(-1)^{k}\Euleriannumber{m-1}{k}</syntaxhighlight> || <math>m \geq 2</math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>BernoulliB[m] == Divide[m,(2)^(m)*((2)^(m)- 1)]*Sum[(- 1)^(k)* Sum[(-1)^m Binomial[m - 1+1,m] (k-m+1)^(m - 1),{m,0,k+1}], {k, 0, m - 2}, GenerateConditions->None]</syntaxhighlight> || Missing Macro Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [1 / 2]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[0.16666666666666666, Times[-0.16666666666666666, NSum[Times[Power[-1, 2], Power[Plus[1, k, Times[-1, 2]], Plus[-1, 2]]]
-Test Values: {2, 0, Plus[1, k]}]]], {Rule[m, 2]}</syntaxhighlight><br></div></div>
-|-
-| [https://dlmf.nist.gov/26.14.E12 26.14.E12] || [[Item:Q7963|<math>\StirlingnumberS@{n}{m} = \frac{1}{m!}\sum_{k=0}^{n-1}\Euleriannumber{n}{k}\binom{k}{n-m}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\StirlingnumberS@{n}{m} = \frac{1}{m!}\sum_{k=0}^{n-1}\Euleriannumber{n}{k}\binom{k}{n-m}</syntaxhighlight> || <math>n \geq m, n \geq 1</math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>StirlingS2[n, m] == Divide[1,(m)!]*Sum[Sum[(-1)^m Binomial[n+1,m] (k-m+1)^(n),{m,0,k+1}]*Binomial[k,n - m], {k, 0, n - 1}, GenerateConditions->None]</syntaxhighlight> || Missing Macro Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [6 / 6]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[1.0, Times[-1.0, NSum[Times[Power[-1, 1], Power[Plus[1, k, Times[-1, 1]], 1], Binomial[Plus[1, 1], 1]]
-Test Values: {1, 0, Plus[1, k]}]]], {Rule[m, 1], Rule[n, 1]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Plus[1.0, Times[-1.0, NSum[Times[Power[-1, 1], Power[Plus[1, k, Times[-1, 1]], 2], Binomial[Plus[1, 2], 1]]
-Test Values: {1, 0, Plus[1, k]}]]], {Rule[m, 1], Rule[n, 2]}</syntaxhighlight><br>... skip entries to safe data</div></div>
-|-
-| [https://dlmf.nist.gov/26.14.E13 26.14.E13] || [[Item:Q7964|<math>\Euleriannumber{0}{k} = \Kroneckerdelta{0}{k}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Euleriannumber{0}{k} = \Kroneckerdelta{0}{k}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sum[(-1)^m Binomial[0+1,m] (k-m+1)^(0),{m,0,k+1}] == KroneckerDelta[0, k]</syntaxhighlight> || Missing Macro Error || Failure || - || Successful [Tested: 3]
-|-
-| [https://dlmf.nist.gov/26.14.E14 26.14.E14] || [[Item:Q7965|<math>\Euleriannumber{n}{0} = 1</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Euleriannumber{n}{0} = 1</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sum[(-1)^m Binomial[n+1,m] (0-m+1)^(n),{m,0,0+1}] == 1</syntaxhighlight> || Missing Macro Error || Failure || - || Successful [Tested: 3]
-|-
-| [https://dlmf.nist.gov/26.14.E15 26.14.E15] || [[Item:Q7966|<math>\Euleriannumber{n}{1} = 2^{n}-n-1</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Euleriannumber{n}{1} = 2^{n}-n-1</syntaxhighlight> || <math>n \geq 1</math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sum[(-1)^m Binomial[n+1,m] (1-m+1)^(n),{m,0,1+1}] == (2)^(n)- n - 1</syntaxhighlight> || Missing Macro Error || Successful || - || Successful [Tested: 3]
-|-
-| [https://dlmf.nist.gov/26.14.E16 26.14.E16] || [[Item:Q7967|<math>\Euleriannumber{n}{2} = 3^{n}-(n+1)2^{n}+\binom{n+1}{2}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Euleriannumber{n}{2} = 3^{n}-(n+1)2^{n}+\binom{n+1}{2}</syntaxhighlight> || <math>n \geq 1</math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sum[(-1)^m Binomial[n+1,m] (2-m+1)^(n),{m,0,2+1}] == (3)^(n)-(n + 1)*(2)^(n)+Binomial[n + 1,2]</syntaxhighlight> || Missing Macro Error || Successful || - || Successful [Tested: 3]
-|-
-| [https://dlmf.nist.gov/26.15.E3 26.15.E3] || [[Item:Q7970|<math>R(x,B) = \sum_{j=0}^{n}r_{j}(B)\,x^{j}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>R(x,B) = \sum_{j=0}^{n}r_{j}(B)\,x^{j}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>R(x , B) = sum(r[j](B)* (x)^(j), j = 0..n)</syntaxhighlight> || <syntaxhighlight lang=mathematica>R[x , B] == Sum[Subscript[r, j][B]* (x)^(j), {j, 0, n}, GenerateConditions->None]</syntaxhighlight> || Skipped - no semantic math || Skipped - no semantic math || - || -
-|-
-| [https://dlmf.nist.gov/26.15.E4 26.15.E4] || [[Item:Q7971|<math>R(x,B) = R(x,B_{1})\,R(x,B_{2})</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>R(x,B) = R(x,B_{1})\,R(x,B_{2})</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>R(x , B) = R(x , B[1])* R(x , B[2])</syntaxhighlight> || <syntaxhighlight lang=mathematica>R[x , B] == R[x , Subscript[B, 1]]* R[x , Subscript[B, 2]]</syntaxhighlight> || Skipped - no semantic math || Skipped - no semantic math || - || -
-|-
-| [https://dlmf.nist.gov/26.15.E6 26.15.E6] || [[Item:Q7973|<math>N(x,B) = \sum_{k=0}^{n}N_{k}(B)\,x^{k}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>N(x,B) = \sum_{k=0}^{n}N_{k}(B)\,x^{k}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>N(x , B) = sum(N[k](B)* (x)^(k), k = 0..n)</syntaxhighlight> || <syntaxhighlight lang=mathematica>N[x , B] == Sum[Subscript[N, k][B]* (x)^(k), {k, 0, n}, GenerateConditions->None]</syntaxhighlight> || Skipped - no semantic math || Skipped - no semantic math || - || -
-|-
-| [https://dlmf.nist.gov/26.15.E7 26.15.E7] || [[Item:Q7974|<math>N(x,B) = \sum_{k=0}^{n}r_{k}(B)(n-k)!(x-1)^{k}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>N(x,B) = \sum_{k=0}^{n}r_{k}(B)(n-k)!(x-1)^{k}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>N(x , B) = sum(r[k](B)*factorial(n - k)*(x - 1)^(k), k = 0..n)</syntaxhighlight> || <syntaxhighlight lang=mathematica>N[x , B] == Sum[Subscript[r, k][B]*(n - k)!*(x - 1)^(k), {k, 0, n}, GenerateConditions->None]</syntaxhighlight> || Skipped - no semantic math || Skipped - no semantic math || - || -
-|-
-| [https://dlmf.nist.gov/26.15.E8 26.15.E8] || [[Item:Q7975|<math>N_{0}(B)\defeq N(0,B) = \sum_{k=0}^{n}(-1)^{k}r_{k}(B)(n-k)!</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>N_{0}(B)\defeq N(0,B) = \sum_{k=0}^{n}(-1)^{k}r_{k}(B)(n-k)!</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>N[0](B) = N(0 , B) = sum((- 1)^(k)* r[k](B)*factorial(n - k), k = 0..n)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Subscript[N, 0][B] == N[0 , B] == Sum[(- 1)^(k)* Subscript[r, k][B]*(n - k)!, {k, 0, n}, GenerateConditions->None]</syntaxhighlight> || Failure || Failure || Error || Error
-|-
-| [https://dlmf.nist.gov/26.15.E9 26.15.E9] || [[Item:Q7976|<math>r_{k}(B) = \frac{2n}{2n-k}\binom{2n-k}{k}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>r_{k}(B) = \frac{2n}{2n-k}\binom{2n-k}{k}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>r[k](B) = (2*n)/(2*n - k)*binomial(2*n - k,k)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Subscript[r, k][B] == Divide[2*n,2*n - k]*Binomial[2*n - k,k]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: -1.500000000+.8660254040*I
-Test Values: {B = 1/2*3^(1/2)+1/2*I, r[k] = 1/2*3^(1/2)+1/2*I, k = 1, n = 1}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -3.500000000+.8660254040*I
-Test Values: {B = 1/2*3^(1/2)+1/2*I, r[k] = 1/2*3^(1/2)+1/2*I, k = 1, n = 2}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[-1.5, 0.8660254037844386]
-Test Values: {Rule[B, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[k, 1], Rule[n, 1], Rule[Subscript[r, k], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[-3.5, 0.8660254037844386]
-Test Values: {Rule[B, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[k, 1], Rule[n, 2], Rule[Subscript[r, k], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
-|-
-| [https://dlmf.nist.gov/26.15.E10 26.15.E10] || [[Item:Q7977|<math>2(n!)N_{0}(B) = 2(n!)\sum_{k=0}^{n}(-1)^{k}\frac{2n}{2n-k}\binom{2n-k}{k}{(n-k)!}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>2(n!)N_{0}(B) = 2(n!)\sum_{k=0}^{n}(-1)^{k}\frac{2n}{2n-k}\binom{2n-k}{k}{(n-k)!}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>2*(factorial(n))*N[0](B) = 2*(factorial(n))*sum((- 1)^(k)*(2*n)/(2*n - k)*binomial(2*n - k,k)*factorial(n - k), k = 0..n)</syntaxhighlight> || <syntaxhighlight lang=mathematica>2*((n)!)*Subscript[N, 0][B] == 2*((n)!)*Sum[(- 1)^(k)*Divide[2*n,2*n - k]*Binomial[2*n - k,k]*(n - k)!, {k, 0, n}, GenerateConditions->None]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [292 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: 3.000000001+1.732050808*I
-Test Values: {B = 1/2*3^(1/2)+1/2*I, N[0] = 1/2*3^(1/2)+1/2*I, n = 1}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: 2.000000002+3.464101616*I
-Test Values: {B = 1/2*3^(1/2)+1/2*I, N[0] = 1/2*3^(1/2)+1/2*I, n = 2}</syntaxhighlight><br>... skip entries to safe data</div></div> || Skipped - Because timed out
-|-
-| [https://dlmf.nist.gov/26.15.E11 26.15.E11] || [[Item:Q7978|<math>\sum_{k=0}^{n}r_{n-k}(B)(x-k+1)_{k} = \prod_{j=1}^{n}(x+b_{j}-j+1)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sum_{k=0}^{n}r_{n-k}(B)(x-k+1)_{k} = \prod_{j=1}^{n}(x+b_{j}-j+1)</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>sum(r[n - k](B)*x - k + 1[k], k = 0..n) = product(x + b[j]- j + 1, j = 1..n)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sum[Subscript[r, n - k][B]*Subscript[x - k + 1, k], {k, 0, n}, GenerateConditions->None] == Product[x + Subscript[b, j]- j + 1, {j, 1, n}, GenerateConditions->None]</syntaxhighlight> || Skipped - no semantic math || Skipped - no semantic math || - || -
-|-
-| [https://dlmf.nist.gov/26.15.E12 26.15.E12] || [[Item:Q7979|<math>\sum_{k=0}^{n}r_{n-k}(B)(x-k+1)_{k} = x^{n}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sum_{k=0}^{n}r_{n-k}(B)(x-k+1)_{k} = x^{n}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>sum(r[n - k](B)*x - k + 1[k], k = 0..n) = (x)^(n)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sum[Subscript[r, n - k][B]*Subscript[x - k + 1, k], {k, 0, n}, GenerateConditions->None] == (x)^(n)</syntaxhighlight> || Skipped - no semantic math || Skipped - no semantic math || - || -
-|-
-| [https://dlmf.nist.gov/26.15.E13 26.15.E13] || [[Item:Q7980|<math>r_{n-k}(B) = \StirlingnumberS@{n}{k}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>r_{n-k}(B) = \StirlingnumberS@{n}{k}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>r[n - k](B) = Stirling2(n, k)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Subscript[r, n - k][B] == StirlingS2[n, k]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: -.4999999996+.8660254040*I
-Test Values: {B = 1/2*3^(1/2)+1/2*I, r[n-k] = 1/2*3^(1/2)+1/2*I, k = 1, n = 1}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -.4999999996+.8660254040*I
-Test Values: {B = 1/2*3^(1/2)+1/2*I, r[n-k] = 1/2*3^(1/2)+1/2*I, k = 1, n = 2}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[-0.4999999999999999, 0.8660254037844386]
-Test Values: {Rule[B, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[k, 1], Rule[n, 1], Rule[Subscript[r, Plus[Times[-1, k], n]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[-0.4999999999999999, 0.8660254037844386]
-Test Values: {Rule[B, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[k, 1], Rule[n, 2], Rule[Subscript[r, Plus[Times[-1, k], n]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
-|-
-|}

Results of Combinatorial Analysis: Difference between revisions

Latest revision as of 17:47, 25 May 2021

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