Bernoulli and Euler Polynomials - 24.15 Related Sequences of Numbers

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
24.15.E1 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \frac{2t}{e^{t}+1} = \sum_{n=1}^{\infty}G_{n}\frac{t^{n}}{n!}}
\frac{2t}{e^{t}+1} = \sum_{n=1}^{\infty}G_{n}\frac{t^{n}}{n!}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
(2*t)/(exp(t)+ 1) = sum(G[n]*((t)^(n))/(factorial(n)), n = 1..infinity)
 
Divide[2*t,Exp[t]+ 1] == Sum[Subscript[G, n]*Divide[(t)^(n),(n)!], {n, 1, Infinity}, GenerateConditions->None]
 Skipped - no semantic math Skipped - no semantic math - -
24.15.E2 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle G_{n} = 2(1-2^{n})\BernoullinumberB{n}}
G_{n} = 2(1-2^{n})\BernoullinumberB{n}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
G[n] = 2*(1 - (2)^(n))*bernoulli(n)

Subscript[G, n] == 2*(1 - (2)^(n))*BernoulliB[n]
Failure Failure
Failed [30 / 30]
Result: -.1339745960+.5000000000*I
Test Values: {G[n] = 1/2*3^(1/2)+1/2*I, n = 1}


Result: 1.866025404+.5000000000*I
Test Values: {G[n] = 1/2*3^(1/2)+1/2*I, n = 2}

... skip entries to safe data
Failed [30 / 30]
Result: Complex[-0.1339745962155613, 0.49999999999999994]
Test Values: {Rule[n, 1], Rule[Subscript[G, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[1.8660254037844388, 0.49999999999999994]
Test Values: {Rule[n, 2], Rule[Subscript[G, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
24.15.E3 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \tan@@{t} = \sum_{n=0}^{\infty}T_{n}\frac{t^{n}}{n!}}
\tan@@{t} = \sum_{n=0}^{\infty}T_{n}\frac{t^{n}}{n!}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
tan(t) = sum(T[n]*((t)^(n))/(factorial(n)), n = 0..infinity)

Tan[t] == Sum[Subscript[T, n]*Divide[(t)^(n),(n)!], {n, 0, Infinity}, GenerateConditions->None]
Failure Failure
Failed [60 / 60]
Result: -14.29465634-.1115650801*I
Test Values: {t = -3/2, T[n] = 1/2*3^(1/2)+1/2*I}


Result: -13.98985487-.1932363871*I
Test Values: {t = -3/2, T[n] = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
Failed [60 / 60]
Result: Complex[-14.29465633421075, -0.1115650800742149]
Test Values: {Rule[t, -1.5], Rule[Subscript[T, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-13.989854867097504, -0.1932363870390304]
Test Values: {Rule[t, -1.5], Rule[Subscript[T, n], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
24.15.E4 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle T_{2n-1} = (-1)^{n-1}\frac{2^{2n}(2^{2n}-1)}{2n}\BernoullinumberB{2n}}
T_{2n-1} = (-1)^{n-1}\frac{2^{2n}(2^{2n}-1)}{2n}\BernoullinumberB{2n}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
T[2*n - 1] = (- 1)^(n - 1)*((2)^(2*n)*((2)^(2*n)- 1))/(2*n)*bernoulli(2*n)

Subscript[T, 2*n - 1] == (- 1)^(n - 1)*Divide[(2)^(2*n)*((2)^(2*n)- 1),2*n]*BernoulliB[2*n]
Failure Failure
Failed [10 / 10]
Result: -15.13397460+.5000000000*I
Test Values: {T[2*n-1] = 1/2*3^(1/2)+1/2*I, n = 3}


Result: -16.50000000+.8660254040*I
Test Values: {T[2*n-1] = -1/2+1/2*I*3^(1/2), n = 3}

... skip entries to safe data
Failed [29 / 30]
Result: Complex[-0.1339745962155613, 0.49999999999999994]
Test Values: {Rule[n, 1], Rule[Subscript[T, Plus[-1, Times[2, n]]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-1.1339745962155612, 0.49999999999999994]
Test Values: {Rule[n, 2], Rule[Subscript[T, Plus[-1, Times[2, n]]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
24.15.E5 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle T_{2n} = 0}
T_{2n} = 0		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
T[2*n] = 0
 
Subscript[T, 2*n] == 0
 Skipped - no semantic math Skipped - no semantic math - -
24.15.E6 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \BernoullinumberB{n} = \sum_{k=0}^{n}(-1)^{k}\frac{k!\StirlingnumberS@{n}{k}}{k+1}}
\BernoullinumberB{n} = \sum_{k=0}^{n}(-1)^{k}\frac{k!\StirlingnumberS@{n}{k}}{k+1}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
bernoulli(n) = sum((- 1)^(k)*(factorial(k)*Stirling2(n, k))/(k + 1), k = 0..n)

BernoulliB[n] == Sum[(- 1)^(k)*Divide[(k)!*StirlingS2[n, k],k + 1], {k, 0, n}, GenerateConditions->None]
Failure Successful Successful [Tested: 3] Successful [Tested: 3]
24.15.E7 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \BernoullinumberB{n} = \sum_{k=0}^{n}(-1)^{k}\binom{n+1}{k+1}\StirlingnumberS@{n+k}{k}\bigg{/}\binom{n+k}{k}}
\BernoullinumberB{n} = \sum_{k=0}^{n}(-1)^{k}\binom{n+1}{k+1}\StirlingnumberS@{n+k}{k}\bigg{/}\binom{n+k}{k}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
bernoulli(n) = sum((- 1)^(k)*binomial(n + 1,k + 1)*Stirling2(n + k, k)/(binomial(n + k,k)), k = 0..n)

BernoulliB[n] == Sum[(- 1)^(k)*Binomial[n + 1,k + 1]*StirlingS2[n + k, k]/(Binomial[n + k,k]), {k, 0, n}, GenerateConditions->None]
Failure Failure Successful [Tested: 3] Successful [Tested: 3]
24.15.E8 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \sum_{k=0}^{n}(-1)^{n+k}\Stirlingnumbers@{n+1}{k+1}\BernoullinumberB{k} = \frac{n!}{n+1}}
\sum_{k=0}^{n}(-1)^{n+k}\Stirlingnumbers@{n+1}{k+1}\BernoullinumberB{k} = \frac{n!}{n+1}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
sum((- 1)^(n + k)* Stirling1(n + 1, k + 1)*bernoulli(k), k = 0..n) = (factorial(n))/(n + 1)

Sum[(- 1)^(n + k)* StirlingS1[n + 1, k + 1]*BernoulliB[k], {k, 0, n}, GenerateConditions->None] == Divide[(n)!,n + 1]
Failure Failure Successful [Tested: 3] Successful [Tested: 3]
24.15.E11 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \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}}
\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}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
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]

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]
Failure Failure
Failed [300 / 300]
Result: .4330127020+.2500000000*I
Test Values: {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}


Result: .4650877169+.2685185185*I
Test Values: {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}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[0.43301270189221935, 0.24999999999999997]
Test Values: {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]]]}


Result: Complex[0.46508771684719863, 0.2685185185185185]
Test Values: {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]]]}

... skip entries to safe data
24.15.E12 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \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}}}
\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}}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
sum(binomial(n,2*k)*((5)/(4))^(k)* euler(2*k)*v[n - 2*k], k = 0..floor((n)/(2))) = (1)/((2)^(n - 1))

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)]
Missing Macro Error Failure -
Failed [29 / 30]
Result: Complex[-0.1339745962155613, 0.49999999999999994]
Test Values: {Rule[n, 1], Rule[Subscript[v, Plus[Times[-2, k], n]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-0.7165063509461097, -0.12499999999999999]
Test Values: {Rule[n, 2], Rule[Subscript[v, Plus[Times[-2, k], n]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
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