24.16

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
24.16.E5	${\displaystyle{\displaystyle\frac{t}{\ln\left(1+t\right)}=\sum_{n=0}^{\infty}b_{n}t^{n}}}$ \frac{t}{\ln@{1+t}} = \sum_{n=0}^{\infty}b_{n}t^{n}	${\displaystyle{\displaystyle|t|<1}}$	(t)/(ln(1 + t)) = sum(b[n]*(t)^(n), n = 0..infinity)	Divide[t,Log[1 + t]] == Sum[Subscript[b, n]*(t)^(n), {n, 0, Infinity}, GenerateConditions->None]	Failure	Failure	Failed [20 / 20] Result: .1439972511-.3333333333*I Test Values: {t = -1/2, b[n] = 1/2*3^(1/2)+1/2*I} Result: 1.054680854-.5773502693*I Test Values: {t = -1/2, b[n] = -1/2+1/2*I*3^(1/2)} ... skip entries to safe data	Failed [20 / 20] Result: Complex[0.14399725125485596, -0.33333333333333326] Test Values: {Rule[t, -0.5], Rule[Subscript[b, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[1.054680853777815, -0.5773502691896257] Test Values: {Rule[t, -0.5], Rule[Subscript[b, n], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
24.16.E7	${\displaystyle{\displaystyle\frac{t}{(1+\lambda t)^{\ifrac{1}{\lambda}}-1}=\sum_{n=0}^{\infty}\beta_{n}(\lambda)\frac{t^{n}}{n!}}}$ \frac{t}{(1+\lambda t)^{\ifrac{1}{\lambda}}-1} = \sum_{n=0}^{\infty}\beta_{n}(\lambda)\frac{t^{n}}{n!}		(t)/((1 + lambda*t)^((1)/(lambda))- 1) = sum(beta[n](lambda)*((t)^(n))/(factorial(n)), n = 0..infinity)	Divide[t,(1 + \[Lambda]*t)^(Divide[1,\[Lambda]])- 1] == Sum[Subscript[\[Beta], n][\[Lambda]]*Divide[(t)^(n),(n)!], {n, 0, Infinity}, GenerateConditions->None]	Skipped - no semantic math	Skipped - no semantic math	-	-
24.16.E8	${\displaystyle{\displaystyle\beta_{n}(\lambda)=n!b_{n}\lambda^{n}+\sum_{k=1}^{\left\lfloor\ifrac{n}{2}\right\rfloor}\frac{n}{2k}B_{2k}s\left(n-1,2k-1\right)\lambda^{n-2k}}}$ \beta_{n}(\lambda) = n!b_{n}\lambda^{n}+\sum_{k=1}^{\floor{\ifrac{n}{2}}}\frac{n}{2k}\BernoullinumberB{2k}\Stirlingnumbers@{n-1}{2k-1}\lambda^{n-2k}		beta[n](lambda) = factorial(n)*b[n]*(lambda)^(n)+ sum((n)/(2*k)*bernoulli(2*k)*Stirling1(n - 1, 2*k - 1)*(lambda)^(n - 2*k), k = 1..floor((n)/(2)))	Subscript[\[Beta], n][\[Lambda]] == (n)!*Subscript[b, n]*\[Lambda]^(n)+ Sum[Divide[n,2*k]*BernoulliB[2*k]*StirlingS1[n - 1, 2*k - 1]*\[Lambda]^(n - 2*k), {k, 1, Floor[Divide[n,2]]}, GenerateConditions->None]	Failure	Failure	Failed [300 / 300] Result: .3333333331-1.133974598*I Test Values: {beta = 3/2, lambda = 1/2*3^(1/2)+1/2*I, b[n] = 1/2*3^(1/2)+1/2*I, beta[n] = 1/2*3^(1/2)+1/2*I, n = 2} Result: -1.032692071-1.500000002*I Test Values: {beta = 3/2, lambda = 1/2*3^(1/2)+1/2*I, b[n] = 1/2*3^(1/2)+1/2*I, beta[n] = -1/2+1/2*I*3^(1/2), n = 2} ... skip entries to safe data	Skipped - Because timed out
24.16.E11	${\displaystyle{\displaystyle B_{n,\chi}(x)=\sum_{k=0}^{n}{n\choose k}B_{k,\chi}x^{n-k}}}$ B_{n,\chi}(x) = \sum_{k=0}^{n}{n\choose k}B_{k,\chi}x^{n-k}		B[n , chi](x) = sum(binomial(n,k)*(B[k , chi](x))^(n - k), k = 0..n)	Subscript[B, n , \[Chi]][x] == Sum[Binomial[n,k]*(Subscript[B, k , \[Chi]][x])^(n - k), {k, 0, n}, GenerateConditions->None]	Skipped - no semantic math	Skipped - no semantic math	-	-
24.16.E12	${\displaystyle{\displaystyle B_{n}\left(x\right)=B_{n,\chi_{0}}(x-1)}}$ \BernoullipolyB{n}@{x} = B_{n,\chi_{0}}(x-1)		bernoulli(n, x) = B[n , chi[0]]*(x - 1)	BernoulliB[n, x] == Subscript[B, n , Subscript[\[Chi], 0]]*(x - 1)	Failure	Failure	Failed [280 / 300] Result: .5669872980-.2500000000*I Test Values: {chi = 1/2*3^(1/2)+1/2*I, x = 3/2, B[n,chi[0]] = 1/2*3^(1/2)+1/2*I, chi[0] = 1/2*3^(1/2)+1/2*I, n = 1} Result: .4836539647-.2500000000*I Test Values: {chi = 1/2*3^(1/2)+1/2*I, x = 3/2, B[n,chi[0]] = 1/2*3^(1/2)+1/2*I, chi[0] = 1/2*3^(1/2)+1/2*I, n = 2} ... skip entries to safe data	Failed [280 / 300] Result: Complex[0.5669872981077806, -0.24999999999999997] Test Values: {Rule[n, 1], Rule[x, 1.5], Rule[χ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[χ, 0], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[B, n, Subscript[χ, 0]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[0.4836539647744474, -0.24999999999999997] Test Values: {Rule[n, 2], Rule[x, 1.5], Rule[χ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[χ, 0], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[B, n, Subscript[χ, 0]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
24.16.E13	${\displaystyle{\displaystyle E_{n}\left(x\right)=-\frac{2^{1-n}}{n+1}B_{n+1,\chi_{4}}(2x-1)}}$ \EulerpolyE{n}@{x} = -\frac{2^{1-n}}{n+1}B_{n+1,\chi_{4}}(2x-1)		euler(n, x) = -((2)^(1 - n))/(n + 1)*B[n + 1 , chi[4]]*(2*x - 1)	EulerE[n, x] == -Divide[(2)^(1 - n),n + 1]*Subscript[B, n + 1 , Subscript[\[Chi], 4]]*(2*x - 1)	Failure	Failure	Failed [290 / 300] Result: 1.866025404+.5000000000*I Test Values: {chi = 1/2*3^(1/2)+1/2*I, x = 3/2, B[n+1,chi[4]] = 1/2*3^(1/2)+1/2*I, chi[4] = 1/2*3^(1/2)+1/2*I, n = 1} Result: 1.038675135+.1666666667*I Test Values: {chi = 1/2*3^(1/2)+1/2*I, x = 3/2, B[n+1,chi[4]] = 1/2*3^(1/2)+1/2*I, chi[4] = 1/2*3^(1/2)+1/2*I, n = 2} ... skip entries to safe data	Failed [290 / 300] Result: Complex[1.8660254037844388, 0.49999999999999994] Test Values: {Rule[n, 1], Rule[x, 1.5], Rule[χ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[χ, 4], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[B, Plus[1, n], Subscript[χ, 4]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[1.0386751345948129, 0.16666666666666663] Test Values: {Rule[n, 2], Rule[x, 1.5], Rule[χ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[χ, 4], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[B, Plus[1, n], Subscript[χ, 4]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data