26.14

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
26.14.E4	${\displaystyle{\displaystyle\sum_{n,k=0}^{\infty}\genfrac{<}{>}{0.0pt}{}{n}{k}x^{k}\,\frac{t^{n}}{n!}=\frac{1-x}{\exp((x-1)t)-x}}}$ \sum_{n,k=0}^{\infty}\Euleriannumber{n}{k}x^{k}\,\frac{t^{n}}{n!} = \frac{1-x}{\exp((x-1)t)-x}	${\displaystyle{\displaystyle|x|<1,|t|<1}}$	Error	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]	Missing Macro Error	Translation Error	-	-
26.14.E5	${\displaystyle{\displaystyle\sum_{k=0}^{n-1}\genfrac{<}{>}{0.0pt}{}{n}{k}\genfrac{(}{)}{0.0pt}{}{x+k}{n}=x^{n}}}$ \sum_{k=0}^{n-1}\Euleriannumber{n}{k}\binom{x+k}{n} = x^{n}		Error	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)	Missing Macro Error	Failure	-	Successful [Tested: 9]
26.14.E6	${\displaystyle{\displaystyle\genfrac{<}{>}{0.0pt}{}{n}{k}=\sum_{j=0}^{k}(-1)^{j}\genfrac{(}{)}{0.0pt}{}{n+1}{j}(k+1-j)^{n}}}$ \Euleriannumber{n}{k} = \sum_{j=0}^{k}(-1)^{j}\binom{n+1}{j}(k+1-j)^{n}	${\displaystyle{\displaystyle n\geq 1}}$	Error	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]	Missing Macro Error	Failure	-	Successful [Tested: 9]
26.14.E7	${\displaystyle{\displaystyle\genfrac{<}{>}{0.0pt}{}{n}{k}=\sum_{j=0}^{n-k}(-1)^{n-k-j}j!\genfrac{(}{)}{0.0pt}{}{n-j}{k}S\left(n,j\right)}}$ \Euleriannumber{n}{k} = \sum_{j=0}^{n-k}(-1)^{n-k-j}j!\binom{n-j}{k}\StirlingnumberS@{n}{j}		Error	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]	Missing Macro Error	Failure	-	Successful [Tested: 9]
26.14.E8	${\displaystyle{\displaystyle\genfrac{<}{>}{0.0pt}{}{n}{k}=(k+1)\genfrac{<}{>}{0.0pt}{}{n-1}{k}+(n-k)\genfrac{<}{>}{0.0pt}{}{n-1}{k-1}}}$ \Euleriannumber{n}{k} = (k+1)\Euleriannumber{n-1}{k}+(n-k)\Euleriannumber{n-1}{k-1}	${\displaystyle{\displaystyle n\geq 2}}$	Error	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}]	Missing Macro Error	Failure	-	Successful [Tested: 6]
26.14.E9	${\displaystyle{\displaystyle\genfrac{<}{>}{0.0pt}{}{n}{k}=\genfrac{<}{>}{0.0pt}{}{n}{n-1-k}}}$ \Euleriannumber{n}{k} = \Euleriannumber{n}{n-1-k}	${\displaystyle{\displaystyle n\geq 1}}$	Error	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}]	Missing Macro Error	Failure	-	Successful [Tested: 9]
26.14.E10	${\displaystyle{\displaystyle\sum_{k=0}^{n-1}\genfrac{<}{>}{0.0pt}{}{n}{k}=n!}}$ \sum_{k=0}^{n-1}\Euleriannumber{n}{k} = n!	${\displaystyle{\displaystyle n\geq 1}}$	Error	Sum[Sum[(-1)^m Binomial[n+1,m] (k-m+1)^(n),{m,0,k+1}], {k, 0, n - 1}, GenerateConditions->None] == (n)!	Missing Macro Error	Failure	-	Successful [Tested: 3]
26.14.E11	${\displaystyle{\displaystyle B_{m}=\frac{m}{2^{m}(2^{m}-1)}\sum_{k=0}^{m-2}(-1)^{k}\genfrac{<}{>}{0.0pt}{}{m-1}{k}}}$ \BernoullinumberB{m} = \frac{m}{2^{m}(2^{m}-1)}\sum_{k=0}^{m-2}(-1)^{k}\Euleriannumber{m-1}{k}	${\displaystyle{\displaystyle m\geq 2}}$	Error	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]	Missing Macro Error	Failure	-	Failed [1 / 2] 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]}
26.14.E12	${\displaystyle{\displaystyle S\left(n,m\right)=\frac{1}{m!}\sum_{k=0}^{n-1}\genfrac{<}{>}{0.0pt}{}{n}{k}\genfrac{(}{)}{0.0pt}{}{k}{n-m}}}$ \StirlingnumberS@{n}{m} = \frac{1}{m!}\sum_{k=0}^{n-1}\Euleriannumber{n}{k}\binom{k}{n-m}	${\displaystyle{\displaystyle n\geq m,n\geq 1}}$	Error	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]	Missing Macro Error	Failure	-	Failed [6 / 6] 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]} 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]} ... skip entries to safe data
26.14.E13	${\displaystyle{\displaystyle\genfrac{<}{>}{0.0pt}{}{0}{k}=\delta_{0,k}}}$ \Euleriannumber{0}{k} = \Kroneckerdelta{0}{k}		Error	Sum[(-1)^m Binomial[0+1,m] (k-m+1)^(0),{m,0,k+1}] == KroneckerDelta[0, k]	Missing Macro Error	Failure	-	Successful [Tested: 3]
26.14.E14	${\displaystyle{\displaystyle\genfrac{<}{>}{0.0pt}{}{n}{0}=1}}$ \Euleriannumber{n}{0} = 1		Error	Sum[(-1)^m Binomial[n+1,m] (0-m+1)^(n),{m,0,0+1}] == 1	Missing Macro Error	Failure	-	Successful [Tested: 3]
26.14.E15	${\displaystyle{\displaystyle\genfrac{<}{>}{0.0pt}{}{n}{1}=2^{n}-n-1}}$ \Euleriannumber{n}{1} = 2^{n}-n-1	${\displaystyle{\displaystyle n\geq 1}}$	Error	Sum[(-1)^m Binomial[n+1,m] (1-m+1)^(n),{m,0,1+1}] == (2)^(n)- n - 1	Missing Macro Error	Successful	-	Successful [Tested: 3]
26.14.E16	${\displaystyle{\displaystyle\genfrac{<}{>}{0.0pt}{}{n}{2}=3^{n}-(n+1)2^{n}+\genfrac{(}{)}{0.0pt}{}{n+1}{2}}}$ \Euleriannumber{n}{2} = 3^{n}-(n+1)2^{n}+\binom{n+1}{2}	${\displaystyle{\displaystyle n\geq 1}}$	Error	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]	Missing Macro Error	Successful	-	Successful [Tested: 3]