Results of Combinatorial Analysis

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DLMF Formula Maple Mathematica Symbolic
Maple		 Symbolic
Mathematica		 Numeric
Maple		 Numeric
Mathematica
26.3.E1 ( m n ) = ( m m - n ) binomial π‘š 𝑛 binomial π‘š π‘š 𝑛 {\displaystyle{\displaystyle\genfrac{(}{)}{0.0pt}{}{m}{n}=\genfrac{(}{)}{0.0pt% }{}{m}{m-n}}} binomial(m,n)=binomial(m,m - n) Binomial[m,n]=Binomial[m,m - n] Failure Successful Skip -
26.3.E1 ( m m - n ) = m ! ( m - n ) ! ⁒ n ! binomial π‘š π‘š 𝑛 π‘š π‘š 𝑛 𝑛 {\displaystyle{\displaystyle\genfrac{(}{)}{0.0pt}{}{m}{m-n}=\frac{m!}{(m-n)!\,% n!}}} binomial(m,m - n)=(factorial(m))/(factorial(m - n)*factorial(n)) Binomial[m,m - n]=Divide[(m)!,(m - n)!*(n)!] Successful Successful - -
26.3.E2 ( m n ) = 0 binomial π‘š 𝑛 0 {\displaystyle{\displaystyle\genfrac{(}{)}{0.0pt}{}{m}{n}=0}} binomial(m,n)= 0 Binomial[m,n]= 0 Failure Failure Skip Successful
26.3.E3 βˆ‘ n = 0 m ( m n ) ⁒ x n = ( 1 + x ) m superscript subscript 𝑛 0 π‘š binomial π‘š 𝑛 superscript π‘₯ 𝑛 superscript 1 π‘₯ π‘š {\displaystyle{\displaystyle\sum_{n=0}^{m}\genfrac{(}{)}{0.0pt}{}{m}{n}x^{n}=(% 1+x)^{m}}} sum(binomial(m,n)*(x)^(n), n = 0..m)=(1 + x)^(m) Sum[Binomial[m,n]*(x)^(n), {n, 0, m}]=(1 + x)^(m) Successful Successful - -
26.3.E4 βˆ‘ m = 0 ∞ ( m + n m ) ⁒ x m = 1 ( 1 - x ) n + 1 superscript subscript π‘š 0 binomial π‘š 𝑛 π‘š superscript π‘₯ π‘š 1 superscript 1 π‘₯ 𝑛 1 {\displaystyle{\displaystyle\sum_{m=0}^{\infty}\genfrac{(}{)}{0.0pt}{}{m+n}{m}% x^{m}=\frac{1}{(1-x)^{n+1}}}} sum(binomial(m + n,m)*(x)^(m), m = 0..infinity)=(1)/((1 - x)^(n + 1)) Sum[Binomial[m + n,m]*(x)^(m), {m, 0, Infinity}]=Divide[1,(1 - x)^(n + 1)] Successful Failure - Successful
26.3.E5 ( m n ) = ( m - 1 n ) + ( m - 1 n - 1 ) binomial π‘š 𝑛 binomial π‘š 1 𝑛 binomial π‘š 1 𝑛 1 {\displaystyle{\displaystyle\genfrac{(}{)}{0.0pt}{}{m}{n}=\genfrac{(}{)}{0.0pt% }{}{m-1}{n}+\genfrac{(}{)}{0.0pt}{}{m-1}{n-1}}} binomial(m,n)=binomial(m - 1,n)+binomial(m - 1,n - 1) Binomial[m,n]=Binomial[m - 1,n]+Binomial[m - 1,n - 1] Successful Successful - -
26.3.E6 ( m n ) = m n ⁒ ( m - 1 n - 1 ) binomial π‘š 𝑛 π‘š 𝑛 binomial π‘š 1 𝑛 1 {\displaystyle{\displaystyle\genfrac{(}{)}{0.0pt}{}{m}{n}=\frac{m}{n}\genfrac{% (}{)}{0.0pt}{}{m-1}{n-1}}} binomial(m,n)=(m)/(n)*binomial(m - 1,n - 1) Binomial[m,n]=Divide[m,n]*Binomial[m - 1,n - 1] Successful Successful - -
26.3.E6 m n ⁒ ( m - 1 n - 1 ) = m - n + 1 n ⁒ ( m n - 1 ) π‘š 𝑛 binomial π‘š 1 𝑛 1 π‘š 𝑛 1 𝑛 binomial π‘š 𝑛 1 {\displaystyle{\displaystyle\frac{m}{n}\genfrac{(}{)}{0.0pt}{}{m-1}{n-1}=\frac% {m-n+1}{n}\genfrac{(}{)}{0.0pt}{}{m}{n-1}}} (m)/(n)*binomial(m - 1,n - 1)=(m - n + 1)/(n)*binomial(m,n - 1) Divide[m,n]*Binomial[m - 1,n - 1]=Divide[m - n + 1,n]*Binomial[m,n - 1] Successful Successful - -
26.3.E7 ( m + 1 n + 1 ) = βˆ‘ k = n m ( k n ) binomial π‘š 1 𝑛 1 superscript subscript π‘˜ 𝑛 π‘š binomial π‘˜ 𝑛 {\displaystyle{\displaystyle\genfrac{(}{)}{0.0pt}{}{m+1}{n+1}=\sum_{k=n}^{m}% \genfrac{(}{)}{0.0pt}{}{k}{n}}} binomial(m + 1,n + 1)= sum(binomial(k,n), k = n..m) Binomial[m + 1,n + 1]= Sum[Binomial[k,n], {k, n, m}] Successful Successful - -
26.3.E8 ( m n ) = βˆ‘ k = 0 n ( m - n - 1 + k k ) binomial π‘š 𝑛 superscript subscript π‘˜ 0 𝑛 binomial π‘š 𝑛 1 π‘˜ π‘˜ {\displaystyle{\displaystyle\genfrac{(}{)}{0.0pt}{}{m}{n}=\sum_{k=0}^{n}% \genfrac{(}{)}{0.0pt}{}{m-n-1+k}{k}}} binomial(m,n)= sum(binomial(m - n - 1 + k,k), k = 0..n) Binomial[m,n]= Sum[Binomial[m - n - 1 + k,k], {k, 0, n}] Successful Successful - -
26.3.E9 ( n 0 ) = ( n n ) binomial 𝑛 0 binomial 𝑛 𝑛 {\displaystyle{\displaystyle\genfrac{(}{)}{0.0pt}{}{n}{0}=\genfrac{(}{)}{0.0pt% }{}{n}{n}}} binomial(n,0)=binomial(n,n) Binomial[n,0]=Binomial[n,n] Successful Successful - -
26.3.E9 ( n n ) = 1 binomial 𝑛 𝑛 1 {\displaystyle{\displaystyle\genfrac{(}{)}{0.0pt}{}{n}{n}=1}} binomial(n,n)= 1 Binomial[n,n]= 1 Successful Successful - -
26.3.E10 ( m n ) = βˆ‘ k = 0 n ( - 1 ) n - k ⁒ ( m + 1 k ) binomial π‘š 𝑛 superscript subscript π‘˜ 0 𝑛 superscript 1 𝑛 π‘˜ binomial π‘š 1 π‘˜ {\displaystyle{\displaystyle\genfrac{(}{)}{0.0pt}{}{m}{n}=\sum_{k=0}^{n}(-1)^{% n-k}\genfrac{(}{)}{0.0pt}{}{m+1}{k}}} binomial(m,n)= sum((- 1)^(n - k)*binomial(m + 1,k), k = 0..n) Binomial[m,n]= Sum[(- 1)^(n - k)*Binomial[m + 1,k], {k, 0, n}] Successful Failure - Successful
26.4.E1 ( n 1 + n 2 n 1 , n 2 ) = ( n 1 + n 2 n 1 ) multinomial subscript 𝑛 1 subscript 𝑛 2 subscript 𝑛 1 subscript 𝑛 2 binomial subscript 𝑛 1 subscript 𝑛 2 subscript 𝑛 1 {\displaystyle{\displaystyle\genfrac{(}{)}{0.0pt}{}{n_{1}+n_{2}}{n_{1},n_{2}}=% \genfrac{(}{)}{0.0pt}{}{n_{1}+n_{2}}{n_{1}}}} multinomial(n[1]+ n[2], n[1], n[2])=binomial(n[1]+ n[2],n[1]) Multinomial[Subscript[n, 1]+ Subscript[n, 2]]=Binomial[Subscript[n, 1]+ Subscript[n, 2],Subscript[n, 1]] Failure Failure Error
Fail
Complex[1.1103428718503567, -2.707788083134227] <- {Rule[Subscript[n, 1], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Subscript[n, 2], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-6.999993813234755, 0.0] <- {Rule[Subscript[n, 1], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Subscript[n, 2], Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[6.885584583718428, -3.3383853150444147] <- {Rule[Subscript[n, 1], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Subscript[n, 2], Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[1.2573870593198788, -0.21521656299188482] <- {Rule[Subscript[n, 1], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Subscript[n, 2], Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
26.4.E1 ( n 1 + n 2 n 1 ) = ( n 1 + n 2 n 2 ) binomial subscript 𝑛 1 subscript 𝑛 2 subscript 𝑛 1 binomial subscript 𝑛 1 subscript 𝑛 2 subscript 𝑛 2 {\displaystyle{\displaystyle\genfrac{(}{)}{0.0pt}{}{n_{1}+n_{2}}{n_{1}}=% \genfrac{(}{)}{0.0pt}{}{n_{1}+n_{2}}{n_{2}}}} binomial(n[1]+ n[2],n[1])=binomial(n[1]+ n[2],n[2]) Binomial[Subscript[n, 1]+ Subscript[n, 2],Subscript[n, 1]]=Binomial[Subscript[n, 1]+ Subscript[n, 2],Subscript[n, 2]] Failure Successful Successful -
26.5.E1 1 n + 1 ⁒ ( 2 ⁒ n n ) = 1 2 ⁒ n + 1 ⁒ ( 2 ⁒ n + 1 n ) 1 𝑛 1 binomial 2 𝑛 𝑛 1 2 𝑛 1 binomial 2 𝑛 1 𝑛 {\displaystyle{\displaystyle\frac{1}{n+1}\genfrac{(}{)}{0.0pt}{}{2n}{n}=\frac{% 1}{2n+1}\genfrac{(}{)}{0.0pt}{}{2n+1}{n}}} (1)/(n + 1)*binomial(2*n,n)=(1)/(2*n + 1)*binomial(2*n + 1,n) Divide[1,n + 1]*Binomial[2*n,n]=Divide[1,2*n + 1]*Binomial[2*n + 1,n] Successful Successful - -
26.5.E1 1 2 ⁒ n + 1 ⁒ ( 2 ⁒ n + 1 n ) = ( 2 ⁒ n n ) - ( 2 ⁒ n n - 1 ) 1 2 𝑛 1 binomial 2 𝑛 1 𝑛 binomial 2 𝑛 𝑛 binomial 2 𝑛 𝑛 1 {\displaystyle{\displaystyle\frac{1}{2n+1}\genfrac{(}{)}{0.0pt}{}{2n+1}{n}=% \genfrac{(}{)}{0.0pt}{}{2n}{n}-\genfrac{(}{)}{0.0pt}{}{2n}{n-1}}} (1)/(2*n + 1)*binomial(2*n + 1,n)=binomial(2*n,n)-binomial(2*n,n - 1) Divide[1,2*n + 1]*Binomial[2*n + 1,n]=Binomial[2*n,n]-Binomial[2*n,n - 1] Successful Failure - Successful
26.5.E1 ( 2 ⁒ n n ) - ( 2 ⁒ n n - 1 ) = ( 2 ⁒ n - 1 n ) - ( 2 ⁒ n - 1 n + 1 ) binomial 2 𝑛 𝑛 binomial 2 𝑛 𝑛 1 binomial 2 𝑛 1 𝑛 binomial 2 𝑛 1 𝑛 1 {\displaystyle{\displaystyle\genfrac{(}{)}{0.0pt}{}{2n}{n}-\genfrac{(}{)}{0.0% pt}{}{2n}{n-1}=\genfrac{(}{)}{0.0pt}{}{2n-1}{n}-\genfrac{(}{)}{0.0pt}{}{2n-1}{% n+1}}} binomial(2*n,n)-binomial(2*n,n - 1)=binomial(2*n - 1,n)-binomial(2*n - 1,n + 1) Binomial[2*n,n]-Binomial[2*n,n - 1]=Binomial[2*n - 1,n]-Binomial[2*n - 1,n + 1] Failure Failure Skip Successful
26.7.E1 B ⁑ ( 0 ) = 1 Bell-number 0 1 {\displaystyle{\displaystyle B\left(0\right)=1}} BellB(0, 1)= 1 BellB[0]= 1 Successful Successful - -
26.7.E2 B ⁑ ( n ) = βˆ‘ k = 0 n S ⁑ ( n , k ) Bell-number 𝑛 superscript subscript π‘˜ 0 𝑛 Stirling-number-second-kind-S 𝑛 π‘˜ {\displaystyle{\displaystyle B\left(n\right)=\sum_{k=0}^{n}S\left(n,k\right)}} BellB(n, 1)= sum(Stirling2(n, k), k = 0..n) BellB[n]= Sum[StirlingS2[n, k], {k, 0, n}] Failure Successful Skip -
26.7.E3 B ⁑ ( n ) = βˆ‘ k = 1 m k n k ! ⁒ βˆ‘ j = 0 m - k ( - 1 ) j j ! Bell-number 𝑛 superscript subscript π‘˜ 1 π‘š superscript π‘˜ 𝑛 π‘˜ superscript subscript 𝑗 0 π‘š π‘˜ superscript 1 𝑗 𝑗 {\displaystyle{\displaystyle B\left(n\right)=\sum_{k=1}^{m}\frac{k^{n}}{k!}% \sum_{j=0}^{m-k}\frac{(-1)^{j}}{j!}}} BellB(n, 1)= sum(((k)^(n))/(factorial(k))*sum(((- 1)^(j))/(factorial(j)), j = 0..m - k), k = 1..m) BellB[n]= Sum[Divide[(k)^(n),(k)!]*Sum[Divide[(- 1)^(j),(j)!], {j, 0, m - k}], {k, 1, m}] Failure Failure Skip Successful
26.7.E4 B ⁑ ( n ) = e - 1 ⁒ βˆ‘ k = 1 ∞ k n k ! Bell-number 𝑛 1 superscript subscript π‘˜ 1 superscript π‘˜ 𝑛 π‘˜ {\displaystyle{\displaystyle B\left(n\right)={\mathrm{e}^{-1}}\sum_{k=1}^{% \infty}\frac{k^{n}}{k!}}} BellB(n, 1)= exp(- 1)*sum(((k)^(n))/(factorial(k)), k = 1..infinity) BellB[n]= Exp[- 1]*Sum[Divide[(k)^(n),(k)!], {k, 1, Infinity}] Failure Failure Skip
Fail
Complex[0.8939534673502062, 0.8939534673502062] <- {Rule[BellB[n], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Sum[Times[Power[k, n], Power[Factorial[k], -1]], {k, 1, DirectedInfinity[1]}], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[0.8939534673502062, 1.934473657395984] <- {Rule[BellB[n], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Sum[Times[Power[k, n], Power[Factorial[k], -1]], {k, 1, DirectedInfinity[1]}], Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[1.934473657395984, 1.934473657395984] <- {Rule[BellB[n], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Sum[Times[Power[k, n], Power[Factorial[k], -1]], {k, 1, DirectedInfinity[1]}], Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[1.934473657395984, 0.8939534673502062] <- {Rule[BellB[n], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Sum[Times[Power[k, n], Power[Factorial[k], -1]], {k, 1, DirectedInfinity[1]}], Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
26.7.E4 e - 1 ⁒ βˆ‘ k = 1 ∞ k n k ! = 1 + ⌊ e - 1 ⁒ βˆ‘ k = 1 2 ⁒ n k n k ! βŒ‹ 1 superscript subscript π‘˜ 1 superscript π‘˜ 𝑛 π‘˜ 1 1 superscript subscript π‘˜ 1 2 𝑛 superscript π‘˜ 𝑛 π‘˜ {\displaystyle{\displaystyle{\mathrm{e}^{-1}}\sum_{k=1}^{\infty}\frac{k^{n}}{k% !}=1+\left\lfloor{\mathrm{e}^{-1}}\sum_{k=1}^{2n}\frac{k^{n}}{k!}\right\rfloor}} exp(- 1)*sum(((k)^(n))/(factorial(k)), k = 1..infinity)= 1 + floor(exp(- 1)*sum(((k)^(n))/(factorial(k)), k = 1..2*n)) Exp[- 1]*Sum[Divide[(k)^(n),(k)!], {k, 1, Infinity}]= 1 + Floor[Exp[- 1]*Sum[Divide[(k)^(n),(k)!], {k, 1, 2*n}]] Failure Failure Skip
Fail
Complex[-0.47973990497711105, 0.520260095022889] <- {Rule[Sum[Times[Power[k, n], Power[Factorial[k], -1]], {k, 1, Times[2, n]}], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Sum[Times[Power[k, n], Power[Factorial[k], -1]], {k, 1, DirectedInfinity[1]}], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-0.47973990497711105, -0.520260095022889] <- {Rule[Sum[Times[Power[k, n], Power[Factorial[k], -1]], {k, 1, Times[2, n]}], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Sum[Times[Power[k, n], Power[Factorial[k], -1]], {k, 1, DirectedInfinity[1]}], Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[-1.520260095022889, -0.520260095022889] <- {Rule[Sum[Times[Power[k, n], Power[Factorial[k], -1]], {k, 1, Times[2, n]}], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Sum[Times[Power[k, n], Power[Factorial[k], -1]], {k, 1, DirectedInfinity[1]}], Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[-1.520260095022889, 0.520260095022889] <- {Rule[Sum[Times[Power[k, n], Power[Factorial[k], -1]], {k, 1, Times[2, n]}], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Sum[Times[Power[k, n], Power[Factorial[k], -1]], {k, 1, DirectedInfinity[1]}], Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
26.7.E5 βˆ‘ n = 0 ∞ B ⁑ ( n ) ⁒ x n n ! = exp ⁑ ( e x - 1 ) superscript subscript 𝑛 0 Bell-number 𝑛 superscript π‘₯ 𝑛 𝑛 π‘₯ 1 {\displaystyle{\displaystyle\sum_{n=0}^{\infty}B\left(n\right)\frac{x^{n}}{n!}% =\exp({\mathrm{e}^{x}}-1)}} sum(BellB(n, 1)*((x)^(n))/(factorial(n)), n = 0..infinity)= exp(exp(x)- 1) Sum[BellB[n]*Divide[(x)^(n),(n)!], {n, 0, Infinity}]= Exp[Exp[x]- 1] Failure Successful Skip -
26.7.E6 B ⁑ ( n + 1 ) = βˆ‘ k = 0 n ( n k ) ⁒ B ⁑ ( k ) Bell-number 𝑛 1 superscript subscript π‘˜ 0 𝑛 binomial 𝑛 π‘˜ Bell-number π‘˜ {\displaystyle{\displaystyle B\left(n+1\right)=\sum_{k=0}^{n}\genfrac{(}{)}{0.% 0pt}{}{n}{k}B\left(k\right)}} BellB(n + 1, 1)= sum(binomial(n,k)*BellB(k, 1), k = 0..n) BellB[n + 1]= Sum[Binomial[n,k]*BellB[k], {k, 0, n}] Failure Failure Skip
Fail
Complex[0.0, 2.8284271247461903] <- {Rule[BellB[Plus[1, n]], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Sum[Times[BellB[k], Binomial[n, k]], {k, 0, n}], Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[2.8284271247461903, 2.8284271247461903] <- {Rule[BellB[Plus[1, n]], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Sum[Times[BellB[k], Binomial[n, k]], {k, 0, n}], Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
2.8284271247461903 <- {Rule[BellB[Plus[1, n]], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Sum[Times[BellB[k], Binomial[n, k]], {k, 0, n}], Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
Complex[0.0, -2.8284271247461903] <- {Rule[BellB[Plus[1, n]], Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[Sum[Times[BellB[k], Binomial[n, k]], {k, 0, n}], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
26.7#Ex1 B ⁑ ( n + 1 ) = βˆ‘ k = 0 n ( n k ) ⁒ B ⁑ ( n ) Bell-number 𝑛 1 superscript subscript π‘˜ 0 𝑛 binomial 𝑛 π‘˜ Bell-number 𝑛 {\displaystyle{\displaystyle B\left(n+1\right)=\sum_{k=0}^{n}\genfrac{(}{)}{0.% 0pt}{}{n}{k}B\left(n\right)}} BellB(n + 1, 1)= sum(binomial(n,k)*BellB(n, 1), k = 0..n) BellB[n + 1]= Sum[Binomial[n,k]*BellB[n], {k, 0, n}] Failure Failure Skip
Fail
-3.0 <- {Rule[n, 2]}
-25.0 <- {Rule[n, 3]}
26.7.E8 N ⁒ ln ⁑ N = n 𝑁 𝑁 𝑛 {\displaystyle{\displaystyle N\ln N=n}} N*ln(N)= n N*Log[N]= n Failure Failure
Fail
-1.130462591+2.090978877*I <- {N = 2^(1/2)+I*2^(1/2), n = 1}
-2.130462591+2.090978877*I <- {N = 2^(1/2)+I*2^(1/2), n = 2}
-3.130462591+2.090978877*I <- {N = 2^(1/2)+I*2^(1/2), n = 3}
-1.130462591-2.090978877*I <- {N = 2^(1/2)-I*2^(1/2), n = 1}
... skip entries to safe data
Fail
Complex[-1.1304625910710442, 2.090978878008139] <- {Rule[n, 1], Rule[N, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-2.130462591071044, 2.090978878008139] <- {Rule[n, 2], Rule[N, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-3.130462591071044, 2.090978878008139] <- {Rule[n, 3], Rule[N, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-1.1304625910710442, -2.090978878008139] <- {Rule[n, 1], Rule[N, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
26.8.E1 s ⁑ ( n , n ) = 1 Stirling-number-first-kind-S 𝑛 𝑛 1 {\displaystyle{\displaystyle s\left(n,n\right)=1}} Stirling1(n, n)= 1 StirlingS1[n, n]= 1 Successful Failure -
Fail
Complex[0.41421356237309515, 1.4142135623730951] <- {Rule[n, 3], Rule[StirlingS1[n, n], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[0.41421356237309515, -1.4142135623730951] <- {Rule[n, 3], Rule[StirlingS1[n, n], Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[-2.414213562373095, -1.4142135623730951] <- {Rule[n, 3], Rule[StirlingS1[n, n], Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[-2.414213562373095, 1.4142135623730951] <- {Rule[n, 3], Rule[StirlingS1[n, n], Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
26.8.E2 s ⁑ ( 1 , k ) = Ξ΄ 1 , k Stirling-number-first-kind-S 1 π‘˜ Kronecker 1 π‘˜ {\displaystyle{\displaystyle s\left(1,k\right)=\delta_{1,k}}} Stirling1(1, k)= KroneckerDelta[1, k] StirlingS1[1, k]= KroneckerDelta[1, k] Failure Failure
Fail
-.414213562-1.414213562*I <- {KroneckerDelta[1,k] = 2^(1/2)+I*2^(1/2), k = 1}
-1.414213562-1.414213562*I <- {KroneckerDelta[1,k] = 2^(1/2)+I*2^(1/2), k = 2}
-1.414213562-1.414213562*I <- {KroneckerDelta[1,k] = 2^(1/2)+I*2^(1/2), k = 3}
-.414213562+1.414213562*I <- {KroneckerDelta[1,k] = 2^(1/2)-I*2^(1/2), k = 1}
... skip entries to safe data
 Successful
26.8.E4 S ⁑ ( n , n ) = 1 Stirling-number-second-kind-S 𝑛 𝑛 1 {\displaystyle{\displaystyle S\left(n,n\right)=1}} Stirling2(n, n)= 1 StirlingS2[n, n]= 1 Successful Failure -
Fail
Complex[0.41421356237309515, 1.4142135623730951] <- {Rule[n, 3], Rule[StirlingS2[n, n], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[0.41421356237309515, -1.4142135623730951] <- {Rule[n, 3], Rule[StirlingS2[n, n], Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[-2.414213562373095, -1.4142135623730951] <- {Rule[n, 3], Rule[StirlingS2[n, n], Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[-2.414213562373095, 1.4142135623730951] <- {Rule[n, 3], Rule[StirlingS2[n, n], Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
26.8.E6 S ⁑ ( n , k ) = 1 k ! ⁒ βˆ‘ j = 0 k ( - 1 ) k - j ⁒ ( k j ) ⁒ j n Stirling-number-second-kind-S 𝑛 π‘˜ 1 π‘˜ superscript subscript 𝑗 0 π‘˜ superscript 1 π‘˜ 𝑗 binomial π‘˜ 𝑗 superscript 𝑗 𝑛 {\displaystyle{\displaystyle S\left(n,k\right)=\frac{1}{k!}\sum_{j=0}^{k}(-1)^% {k-j}\genfrac{(}{)}{0.0pt}{}{k}{j}j^{n}}} Stirling2(n, k)=(1)/(factorial(k))*sum((- 1)^(k - j)*binomial(k,j)*(j)^(n), j = 0..k) StirlingS2[n, k]=Divide[1,(k)!]*Sum[(- 1)^(k - j)*Binomial[k,j]*(j)^(n), {j, 0, k}] Failure Failure Skip Successful
26.8.E7 βˆ‘ k = 0 n s ⁑ ( n , k ) ⁒ x k = ( x - n + 1 ) n superscript subscript π‘˜ 0 𝑛 Stirling-number-first-kind-S 𝑛 π‘˜ superscript π‘₯ π‘˜ subscript π‘₯ 𝑛 1 𝑛 {\displaystyle{\displaystyle\sum_{k=0}^{n}s\left(n,k\right)x^{k}=(x-n+1)_{n}}} sum(Stirling1(n, k)*(x)^(k), k = 0..n)=x - n + 1[n] Sum[StirlingS1[n, k]*(x)^(k), {k, 0, n}]=Subscript[x - n + 1, n] Failure Failure Skip Successful
26.8.E8 βˆ‘ n = 0 ∞ s ⁑ ( n , k ) ⁒ x n n ! = ( ln ⁑ ( 1 + x ) ) k k ! superscript subscript 𝑛 0 Stirling-number-first-kind-S 𝑛 π‘˜ superscript π‘₯ 𝑛 𝑛 superscript 1 π‘₯ π‘˜ π‘˜ {\displaystyle{\displaystyle\sum_{n=0}^{\infty}s\left(n,k\right)\frac{x^{n}}{n% !}=\frac{(\ln\left(1+x\right))^{k}}{k!}}} sum(Stirling1(n, k)*((x)^(n))/(factorial(n)), n = 0..infinity)=((ln(1 + x))^(k))/(factorial(k)) Sum[StirlingS1[n, k]*Divide[(x)^(n),(n)!], {n, 0, Infinity}]=Divide[(Log[1 + x])^(k),(k)!] Failure Failure Skip Skip
26.8.E9 βˆ‘ n , k = 0 ∞ s ⁑ ( n , k ) ⁒ x n n ! ⁒ y k = ( 1 + x ) y superscript subscript 𝑛 π‘˜ 0 Stirling-number-first-kind-S 𝑛 π‘˜ superscript π‘₯ 𝑛 𝑛 superscript 𝑦 π‘˜ superscript 1 π‘₯ 𝑦 {\displaystyle{\displaystyle\sum_{n,k=0}^{\infty}s\left(n,k\right)\frac{x^{n}}% {n!}y^{k}=(1+x)^{y}}} sum(sum(Stirling1(n, k)*((x)^(n))/(factorial(n))*(y)^(k), k = 0..infinity), n = 0..infinity)=(1 + x)^(y) Sum[Sum[StirlingS1[n, k]*Divide[(x)^(n),(n)!]*(y)^(k), {k, 0, Infinity}], {n, 0, Infinity}]=(1 + x)^(y) Failure Failure Skip Skip
26.8.E10 βˆ‘ k = 1 n S ⁑ ( n , k ) ⁒ ( x - k + 1 ) k = x n superscript subscript π‘˜ 1 𝑛 Stirling-number-second-kind-S 𝑛 π‘˜ subscript π‘₯ π‘˜ 1 π‘˜ superscript π‘₯ 𝑛 {\displaystyle{\displaystyle\sum_{k=1}^{n}S\left(n,k\right)(x-k+1)_{k}=x^{n}}} sum(Stirling2(n, k)*x - k + 1[k], k = 1..n)= (x)^(n) Sum[StirlingS2[n, k]*Subscript[x - k + 1, k], {k, 1, n}]= (x)^(n) Failure Failure Skip Successful
26.8.E12 βˆ‘ n = 0 ∞ S ⁑ ( n , k ) ⁒ x n n ! = ( e x - 1 ) k k ! superscript subscript 𝑛 0 Stirling-number-second-kind-S 𝑛 π‘˜ superscript π‘₯ 𝑛 𝑛 superscript π‘₯ 1 π‘˜ π‘˜ {\displaystyle{\displaystyle\sum_{n=0}^{\infty}S\left(n,k\right)\frac{x^{n}}{n% !}=\frac{({\mathrm{e}^{x}}-1)^{k}}{k!}}} sum(Stirling2(n, k)*((x)^(n))/(factorial(n)), n = 0..infinity)=((exp(x)- 1)^(k))/(factorial(k)) Sum[StirlingS2[n, k]*Divide[(x)^(n),(n)!], {n, 0, Infinity}]=Divide[(Exp[x]- 1)^(k),(k)!] Failure Failure Skip Skip
26.8.E13 βˆ‘ n , k = 0 ∞ S ⁑ ( n , k ) ⁒ x n n ! ⁒ y k = exp ⁑ ( y ⁒ ( e x - 1 ) ) superscript subscript 𝑛 π‘˜ 0 Stirling-number-second-kind-S 𝑛 π‘˜ superscript π‘₯ 𝑛 𝑛 superscript 𝑦 π‘˜ 𝑦 π‘₯ 1 {\displaystyle{\displaystyle\sum_{n,k=0}^{\infty}S\left(n,k\right)\frac{x^{n}}% {n!}y^{k}=\exp\left(y({\mathrm{e}^{x}}-1)\right)}} sum(sum(Stirling2(n, k)*((x)^(n))/(factorial(n))*(y)^(k), k = 0..infinity), n = 0..infinity)= exp(y*(exp(x)- 1)) Sum[Sum[StirlingS2[n, k]*Divide[(x)^(n),(n)!]*(y)^(k), {k, 0, Infinity}], {n, 0, Infinity}]= Exp[y*(Exp[x]- 1)] Failure Failure Skip Skip
26.8#Ex1 s ⁑ ( n , 0 ) = 0 Stirling-number-first-kind-S 𝑛 0 0 {\displaystyle{\displaystyle s\left(n,0\right)=0}} Stirling1(n, 0)= 0 StirlingS1[n, 0]= 0 Failure Failure Successful
Fail
Complex[1.4142135623730951, 1.4142135623730951] <- {Rule[StirlingS1[n, 0], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[1.4142135623730951, -1.4142135623730951] <- {Rule[StirlingS1[n, 0], Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[-1.4142135623730951, -1.4142135623730951] <- {Rule[StirlingS1[n, 0], Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[-1.4142135623730951, 1.4142135623730951] <- {Rule[StirlingS1[n, 0], Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
26.8#Ex2 s ⁑ ( n , 1 ) = ( - 1 ) n - 1 ⁒ ( n - 1 ) ! Stirling-number-first-kind-S 𝑛 1 superscript 1 𝑛 1 𝑛 1 {\displaystyle{\displaystyle s\left(n,1\right)=(-1)^{n-1}(n-1)!}} Stirling1(n, 1)=(- 1)^(n - 1)*factorial(n - 1) StirlingS1[n, 1]=(- 1)^(n - 1)*(n - 1)! Failure Failure Successful Successful
26.8.E16 - s ⁑ ( n , n - 1 ) = S ⁑ ( n , n - 1 ) Stirling-number-first-kind-S 𝑛 𝑛 1 Stirling-number-second-kind-S 𝑛 𝑛 1 {\displaystyle{\displaystyle-s\left(n,n-1\right)=S\left(n,n-1\right)}} - Stirling1(n, n - 1)= Stirling2(n, n - 1) - StirlingS1[n, n - 1]= StirlingS2[n, n - 1] Successful Failure -
Fail
Complex[-2.8284271247461903, -2.8284271247461903] <- {Rule[StirlingS1[n, Plus[-1, n]], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[StirlingS2[n, Plus[-1, n]], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
-2.8284271247461903 <- {Rule[StirlingS1[n, Plus[-1, n]], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[StirlingS2[n, Plus[-1, n]], Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[0.0, -2.8284271247461903] <- {Rule[StirlingS1[n, Plus[-1, n]], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[StirlingS2[n, Plus[-1, n]], Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
-2.8284271247461903 <- {Rule[StirlingS1[n, Plus[-1, n]], Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[StirlingS2[n, Plus[-1, n]], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
26.8.E16 S ⁑ ( n , n - 1 ) = ( n 2 ) Stirling-number-second-kind-S 𝑛 𝑛 1 binomial 𝑛 2 {\displaystyle{\displaystyle S\left(n,n-1\right)=\genfrac{(}{)}{0.0pt}{}{n}{2}}} Stirling2(n, n - 1)=binomial(n,2) StirlingS2[n, n - 1]=Binomial[n,2] Successful Failure - Successful
26.8#Ex3 S ⁑ ( n , 0 ) = 0 Stirling-number-second-kind-S 𝑛 0 0 {\displaystyle{\displaystyle S\left(n,0\right)=0}} Stirling2(n, 0)= 0 StirlingS2[n, 0]= 0 Failure Failure Successful
Fail
Complex[1.4142135623730951, 1.4142135623730951] <- {Rule[StirlingS2[n, 0], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[1.4142135623730951, -1.4142135623730951] <- {Rule[StirlingS2[n, 0], Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[-1.4142135623730951, -1.4142135623730951] <- {Rule[StirlingS2[n, 0], Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[-1.4142135623730951, 1.4142135623730951] <- {Rule[StirlingS2[n, 0], Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
26.8#Ex4 S ⁑ ( n , 1 ) = 1 Stirling-number-second-kind-S 𝑛 1 1 {\displaystyle{\displaystyle S\left(n,1\right)=1}} Stirling2(n, 1)= 1 StirlingS2[n, 1]= 1 Failure Failure Successful
Fail
Complex[0.41421356237309515, 1.4142135623730951] <- {Rule[StirlingS2[n, 1], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[0.41421356237309515, -1.4142135623730951] <- {Rule[StirlingS2[n, 1], Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[-2.414213562373095, -1.4142135623730951] <- {Rule[StirlingS2[n, 1], Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[-2.414213562373095, 1.4142135623730951] <- {Rule[StirlingS2[n, 1], Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
26.8#Ex5 S ⁑ ( n , 2 ) = 2 n - 1 - 1 Stirling-number-second-kind-S 𝑛 2 superscript 2 𝑛 1 1 {\displaystyle{\displaystyle S\left(n,2\right)=2^{n-1}-1}} Stirling2(n, 2)= (2)^(n - 1)- 1 StirlingS2[n, 2]= (2)^(n - 1)- 1 Failure Failure Successful Successful
26.8.E18 s ⁑ ( n , k ) = s ⁑ ( n - 1 , k - 1 ) - ( n - 1 ) ⁒ s ⁑ ( n - 1 , k ) Stirling-number-first-kind-S 𝑛 π‘˜ Stirling-number-first-kind-S 𝑛 1 π‘˜ 1 𝑛 1 Stirling-number-first-kind-S 𝑛 1 π‘˜ {\displaystyle{\displaystyle s\left(n,k\right)=s\left(n-1,k-1\right)-(n-1)s% \left(n-1,k\right)}} Stirling1(n, k)= Stirling1(n - 1, k - 1)-(n - 1)* Stirling1(n - 1, k) StirlingS1[n, k]= StirlingS1[n - 1, k - 1]-(n - 1)* StirlingS1[n - 1, k] Failure Failure Successful Successful
26.8.E19 ( k h ) ⁒ s ⁑ ( n , k ) = βˆ‘ j = k - h n - h ( n j ) ⁒ s ⁑ ( n - j , h ) ⁒ s ⁑ ( j , k - h ) binomial π‘˜ β„Ž Stirling-number-first-kind-S 𝑛 π‘˜ superscript subscript 𝑗 π‘˜ β„Ž 𝑛 β„Ž binomial 𝑛 𝑗 Stirling-number-first-kind-S 𝑛 𝑗 β„Ž Stirling-number-first-kind-S 𝑗 π‘˜ β„Ž {\displaystyle{\displaystyle\genfrac{(}{)}{0.0pt}{}{k}{h}s\left(n,k\right)=% \sum_{j=k-h}^{n-h}\genfrac{(}{)}{0.0pt}{}{n}{j}s\left(n-j,h\right)s\left(j,k-h% \right)}} binomial(k,h)*Stirling1(n, k)= sum(binomial(n,j)*Stirling1(n - j, h)*Stirling1(j, k - h), j = k - h..n - h) Binomial[k,h]*StirlingS1[n, k]= Sum[Binomial[n,j]*StirlingS1[n - j, h]*StirlingS1[j, k - h], {j, k - h, n - h}] Failure Failure Skip Successful
26.8.E20 s ⁑ ( n + 1 , k + 1 ) = n ! ⁒ βˆ‘ j = k n ( - 1 ) n - j j ! ⁒ s ⁑ ( j , k ) Stirling-number-first-kind-S 𝑛 1 π‘˜ 1 𝑛 superscript subscript 𝑗 π‘˜ 𝑛 superscript 1 𝑛 𝑗 𝑗 Stirling-number-first-kind-S 𝑗 π‘˜ {\displaystyle{\displaystyle s\left(n+1,k+1\right)=n!\sum_{j=k}^{n}\frac{(-1)^% {n-j}}{j!}\,s\left(j,k\right)}} Stirling1(n + 1, k + 1)= factorial(n)*sum(((- 1)^(n - j))/(factorial(j))*Stirling1(j, k), j = k..n) StirlingS1[n + 1, k + 1]= (n)!*Sum[Divide[(- 1)^(n - j),(j)!]*StirlingS1[j, k], {j, k, n}] Failure Failure Skip Successful
26.8.E21 s ⁑ ( n + k + 1 , k ) = - βˆ‘ j = 0 k ( n + j ) ⁒ s ⁑ ( n + j , j ) Stirling-number-first-kind-S 𝑛 π‘˜ 1 π‘˜ superscript subscript 𝑗 0 π‘˜ 𝑛 𝑗 Stirling-number-first-kind-S 𝑛 𝑗 𝑗 {\displaystyle{\displaystyle s\left(n+k+1,k\right)=-\sum_{j=0}^{k}(n+j)s\left(% n+j,j\right)}} Stirling1(n + k + 1, k)= - sum((n + j)* Stirling1(n + j, j), j = 0..k) StirlingS1[n + k + 1, k]= - Sum[(n + j)* StirlingS1[n + j, j], {j, 0, k}] Failure Failure Skip
Fail
Complex[2.8284271247461903, 2.8284271247461903] <- {Rule[StirlingS1[Plus[1, k, n], k], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Sum[Times[Plus[j, n], StirlingS1[Plus[j, n], j]], {j, 0, k}], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
2.8284271247461903 <- {Rule[StirlingS1[Plus[1, k, n], k], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Sum[Times[Plus[j, n], StirlingS1[Plus[j, n], j]], {j, 0, k}], Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[0.0, 2.8284271247461903] <- {Rule[StirlingS1[Plus[1, k, n], k], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Sum[Times[Plus[j, n], StirlingS1[Plus[j, n], j]], {j, 0, k}], Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
2.8284271247461903 <- {Rule[StirlingS1[Plus[1, k, n], k], Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[Sum[Times[Plus[j, n], StirlingS1[Plus[j, n], j]], {j, 0, k}], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
26.8.E22 S ⁑ ( n , k ) = k ⁒ S ⁑ ( n - 1 , k ) + S ⁑ ( n - 1 , k - 1 ) Stirling-number-second-kind-S 𝑛 π‘˜ π‘˜ Stirling-number-second-kind-S 𝑛 1 π‘˜ Stirling-number-second-kind-S 𝑛 1 π‘˜ 1 {\displaystyle{\displaystyle S\left(n,k\right)=kS\left(n-1,k\right)+S\left(n-1% ,k-1\right)}} Stirling2(n, k)= k*Stirling2(n - 1, k)+ Stirling2(n - 1, k - 1) StirlingS2[n, k]= k*StirlingS2[n - 1, k]+ StirlingS2[n - 1, k - 1] Failure Failure Successful Successful
26.8.E23 ( k h ) ⁒ S ⁑ ( n , k ) = βˆ‘ j = k - h n - h ( n j ) ⁒ S ⁑ ( n - j , h ) ⁒ S ⁑ ( j , k - h ) binomial π‘˜ β„Ž Stirling-number-second-kind-S 𝑛 π‘˜ superscript subscript 𝑗 π‘˜ β„Ž 𝑛 β„Ž binomial 𝑛 𝑗 Stirling-number-second-kind-S 𝑛 𝑗 β„Ž Stirling-number-second-kind-S 𝑗 π‘˜ β„Ž {\displaystyle{\displaystyle\genfrac{(}{)}{0.0pt}{}{k}{h}S\left(n,k\right)=% \sum_{j=k-h}^{n-h}\genfrac{(}{)}{0.0pt}{}{n}{j}S\left(n-j,h\right)S\left(j,k-h% \right)}} binomial(k,h)*Stirling2(n, k)= sum(binomial(n,j)*Stirling2(n - j, h)*Stirling2(j, k - h), j = k - h..n - h) Binomial[k,h]*StirlingS2[n, k]= Sum[Binomial[n,j]*StirlingS2[n - j, h]*StirlingS2[j, k - h], {j, k - h, n - h}] Failure Failure Skip Successful
26.8.E24 S ⁑ ( n , k ) = βˆ‘ j = k n S ⁑ ( j - 1 , k - 1 ) ⁒ k n - j Stirling-number-second-kind-S 𝑛 π‘˜ superscript subscript 𝑗 π‘˜ 𝑛 Stirling-number-second-kind-S 𝑗 1 π‘˜ 1 superscript π‘˜ 𝑛 𝑗 {\displaystyle{\displaystyle S\left(n,k\right)=\sum_{j=k}^{n}S\left(j-1,k-1% \right)k^{n-j}}} Stirling2(n, k)= sum(Stirling2(j - 1, k - 1)*(k)^(n - j), j = k..n) StirlingS2[n, k]= Sum[StirlingS2[j - 1, k - 1]*(k)^(n - j), {j, k, n}] Failure Failure Skip
Fail
Complex[0.0, 2.8284271247461903] <- {Rule[StirlingS2[n, k], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Sum[Times[Power[k, Plus[Times[-1, j], n]], StirlingS2[Plus[-1, j], Plus[-1, k]]], {j, k, n}], Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[2.8284271247461903, 2.8284271247461903] <- {Rule[StirlingS2[n, k], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Sum[Times[Power[k, Plus[Times[-1, j], n]], StirlingS2[Plus[-1, j], Plus[-1, k]]], {j, k, n}], Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
2.8284271247461903 <- {Rule[StirlingS2[n, k], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Sum[Times[Power[k, Plus[Times[-1, j], n]], StirlingS2[Plus[-1, j], Plus[-1, k]]], {j, k, n}], Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
Complex[0.0, -2.8284271247461903] <- {Rule[StirlingS2[n, k], Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[Sum[Times[Power[k, Plus[Times[-1, j], n]], StirlingS2[Plus[-1, j], Plus[-1, k]]], {j, k, n}], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
26.8.E25 S ⁑ ( n + 1 , k + 1 ) = βˆ‘ j = k n ( n j ) ⁒ S ⁑ ( j , k ) Stirling-number-second-kind-S 𝑛 1 π‘˜ 1 superscript subscript 𝑗 π‘˜ 𝑛 binomial 𝑛 𝑗 Stirling-number-second-kind-S 𝑗 π‘˜ {\displaystyle{\displaystyle S\left(n+1,k+1\right)=\sum_{j=k}^{n}\genfrac{(}{)% }{0.0pt}{}{n}{j}S\left(j,k\right)}} Stirling2(n + 1, k + 1)= sum(binomial(n,j)*Stirling2(j, k), j = k..n) StirlingS2[n + 1, k + 1]= Sum[Binomial[n,j]*StirlingS2[j, k], {j, k, n}] Failure Failure Skip
Fail
Complex[0.0, 2.8284271247461903] <- {Rule[StirlingS2[Plus[1, n], Plus[1, k]], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Sum[Times[Binomial[n, j], StirlingS2[j, k]], {j, k, n}], Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[2.8284271247461903, 2.8284271247461903] <- {Rule[StirlingS2[Plus[1, n], Plus[1, k]], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Sum[Times[Binomial[n, j], StirlingS2[j, k]], {j, k, n}], Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
2.8284271247461903 <- {Rule[StirlingS2[Plus[1, n], Plus[1, k]], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Sum[Times[Binomial[n, j], StirlingS2[j, k]], {j, k, n}], Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
Complex[0.0, -2.8284271247461903] <- {Rule[StirlingS2[Plus[1, n], Plus[1, k]], Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[Sum[Times[Binomial[n, j], StirlingS2[j, k]], {j, k, n}], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
26.8.E26 S ⁑ ( n + k + 1 , k ) = βˆ‘ j = 0 k j ⁒ S ⁑ ( n + j , j ) Stirling-number-second-kind-S 𝑛 π‘˜ 1 π‘˜ superscript subscript 𝑗 0 π‘˜ 𝑗 Stirling-number-second-kind-S 𝑛 𝑗 𝑗 {\displaystyle{\displaystyle S\left(n+k+1,k\right)=\sum_{j=0}^{k}jS\left(n+j,j% \right)}} Stirling2(n + k + 1, k)= sum(j*Stirling2(n + j, j), j = 0..k) StirlingS2[n + k + 1, k]= Sum[j*StirlingS2[n + j, j], {j, 0, k}] Failure Successful Skip -
26.8.E27 s ⁑ ( n , n - k ) = βˆ‘ j = 0 k ( - 1 ) j ⁒ ( n - 1 + j k + j ) ⁒ ( n + k k - j ) ⁒ S ⁑ ( k + j , j ) Stirling-number-first-kind-S 𝑛 𝑛 π‘˜ superscript subscript 𝑗 0 π‘˜ superscript 1 𝑗 binomial 𝑛 1 𝑗 π‘˜ 𝑗 binomial 𝑛 π‘˜ π‘˜ 𝑗 Stirling-number-second-kind-S π‘˜ 𝑗 𝑗 {\displaystyle{\displaystyle s\left(n,n-k\right)=\sum_{j=0}^{k}(-1)^{j}% \genfrac{(}{)}{0.0pt}{}{n-1+j}{k+j}\,\genfrac{(}{)}{0.0pt}{}{n+k}{k-j}\*S\left% (k+j,j\right)}} 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) 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}] Failure Failure Skip
Fail
Complex[0.0, 2.8284271247461903] <- {Rule[StirlingS1[n, Plus[Times[-1, k], n]], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Sum[Times[Power[-1, j], Binomial[Plus[-1, j, n], Plus[j, k]], Binomial[Plus[k, n], Plus[Times[-1, j], k]], StirlingS2[Plus[j, k], j]], {j, 0, k}], Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[2.8284271247461903, 2.8284271247461903] <- {Rule[StirlingS1[n, Plus[Times[-1, k], n]], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Sum[Times[Power[-1, j], Binomial[Plus[-1, j, n], Plus[j, k]], Binomial[Plus[k, n], Plus[Times[-1, j], k]], StirlingS2[Plus[j, k], j]], {j, 0, k}], Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
2.8284271247461903 <- {Rule[StirlingS1[n, Plus[Times[-1, k], n]], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Sum[Times[Power[-1, j], Binomial[Plus[-1, j, n], Plus[j, k]], Binomial[Plus[k, n], Plus[Times[-1, j], k]], StirlingS2[Plus[j, k], j]], {j, 0, k}], Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
Complex[0.0, -2.8284271247461903] <- {Rule[StirlingS1[n, Plus[Times[-1, k], n]], Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[Sum[Times[Power[-1, j], Binomial[Plus[-1, j, n], Plus[j, k]], Binomial[Plus[k, n], Plus[Times[-1, j], k]], StirlingS2[Plus[j, k], j]], {j, 0, k}], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
26.8.E28 βˆ‘ k = 1 n s ⁑ ( n , k ) = 0 superscript subscript π‘˜ 1 𝑛 Stirling-number-first-kind-S 𝑛 π‘˜ 0 {\displaystyle{\displaystyle\sum_{k=1}^{n}s\left(n,k\right)=0}} sum(Stirling1(n, k), k = 1..n)= 0 Sum[StirlingS1[n, k], {k, 1, n}]= 0 Failure Failure Skip
Fail
Complex[1.4142135623730951, 1.4142135623730951] <- {Rule[n, Rational[3, 2]], Rule[Sum[StirlingS1[n, k], {k, 1, n}], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[1.4142135623730951, -1.4142135623730951] <- {Rule[n, Rational[3, 2]], Rule[Sum[StirlingS1[n, k], {k, 1, n}], Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[-1.4142135623730951, -1.4142135623730951] <- {Rule[n, Rational[3, 2]], Rule[Sum[StirlingS1[n, k], {k, 1, n}], Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[-1.4142135623730951, 1.4142135623730951] <- {Rule[n, Rational[3, 2]], Rule[Sum[StirlingS1[n, k], {k, 1, n}], Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
26.8.E29 βˆ‘ k = 1 n ( - 1 ) n - k ⁒ s ⁑ ( n , k ) = n ! superscript subscript π‘˜ 1 𝑛 superscript 1 𝑛 π‘˜ Stirling-number-first-kind-S 𝑛 π‘˜ 𝑛 {\displaystyle{\displaystyle\sum_{k=1}^{n}(-1)^{n-k}s\left(n,k\right)=n!}} sum((- 1)^(n - k)* Stirling1(n, k), k = 1..n)= factorial(n) Sum[(- 1)^(n - k)* StirlingS1[n, k], {k, 1, n}]= (n)! Failure Failure Skip
Fail
-2.0 <- {Rule[n, 2]}
-7.0 <- {Rule[n, 3]}
26.8.E30 βˆ‘ j = k n s ⁑ ( n + 1 , j + 1 ) ⁒ n j - k = s ⁑ ( n , k ) superscript subscript 𝑗 π‘˜ 𝑛 Stirling-number-first-kind-S 𝑛 1 𝑗 1 superscript 𝑛 𝑗 π‘˜ Stirling-number-first-kind-S 𝑛 π‘˜ {\displaystyle{\displaystyle\sum_{j=k}^{n}s\left(n+1,j+1\right)\,n^{j-k}=s% \left(n,k\right)}} sum(Stirling1(n + 1, j + 1)*(n)^(j - k), j = k..n)= Stirling1(n, k) Sum[StirlingS1[n + 1, j + 1]*(n)^(j - k), {j, k, n}]= StirlingS1[n, k] Failure Successful Skip -
26.8.E33 S ⁑ ( n , n - k ) = βˆ‘ j = 0 k ( - 1 ) j ⁒ ( n - 1 + j k + j ) ⁒ ( n + k k - j ) ⁒ s ⁑ ( k + j , j ) Stirling-number-second-kind-S 𝑛 𝑛 π‘˜ superscript subscript 𝑗 0 π‘˜ superscript 1 𝑗 binomial 𝑛 1 𝑗 π‘˜ 𝑗 binomial 𝑛 π‘˜ π‘˜ 𝑗 Stirling-number-first-kind-S π‘˜ 𝑗 𝑗 {\displaystyle{\displaystyle S\left(n,n-k\right)=\sum_{j=0}^{k}(-1)^{j}% \genfrac{(}{)}{0.0pt}{}{n-1+j}{k+j}\genfrac{(}{)}{0.0pt}{}{n+k}{k-j}\*s\left(k% +j,j\right)}} 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) 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}] Failure Failure Skip
Fail
Complex[0.0, 2.8284271247461903] <- {Rule[StirlingS2[n, Plus[Times[-1, k], n]], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Sum[Times[Power[-1, j], Binomial[Plus[-1, j, n], Plus[j, k]], Binomial[Plus[k, n], Plus[Times[-1, j], k]], StirlingS1[Plus[j, k], j]], {j, 0, k}], Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[2.8284271247461903, 2.8284271247461903] <- {Rule[StirlingS2[n, Plus[Times[-1, k], n]], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Sum[Times[Power[-1, j], Binomial[Plus[-1, j, n], Plus[j, k]], Binomial[Plus[k, n], Plus[Times[-1, j], k]], StirlingS1[Plus[j, k], j]], {j, 0, k}], Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
2.8284271247461903 <- {Rule[StirlingS2[n, Plus[Times[-1, k], n]], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Sum[Times[Power[-1, j], Binomial[Plus[-1, j, n], Plus[j, k]], Binomial[Plus[k, n], Plus[Times[-1, j], k]], StirlingS1[Plus[j, k], j]], {j, 0, k}], Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
Complex[0.0, -2.8284271247461903] <- {Rule[StirlingS2[n, Plus[Times[-1, k], n]], Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[Sum[Times[Power[-1, j], Binomial[Plus[-1, j, n], Plus[j, k]], Binomial[Plus[k, n], Plus[Times[-1, j], k]], StirlingS1[Plus[j, k], j]], {j, 0, k}], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
26.8.E34 βˆ‘ j = 0 n j k ⁒ x j = βˆ‘ j = 0 k S ⁑ ( k , j ) ⁒ x j ⁒ d j d x j ⁑ ( 1 - x n + 1 1 - x ) superscript subscript 𝑗 0 𝑛 superscript 𝑗 π‘˜ superscript π‘₯ 𝑗 superscript subscript 𝑗 0 π‘˜ Stirling-number-second-kind-S π‘˜ 𝑗 superscript π‘₯ 𝑗 derivative π‘₯ 𝑗 1 superscript π‘₯ 𝑛 1 1 π‘₯ {\displaystyle{\displaystyle\sum_{j=0}^{n}j^{k}x^{j}=\sum_{j=0}^{k}S\left(k,j% \right)x^{j}\frac{{\mathrm{d}}^{j}}{{\mathrm{d}x}^{j}}\left(\frac{1-x^{n+1}}{1% -x}\right)}} 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) Sum[(j)^(k)* (x)^(j), {j, 0, n}]= Sum[StirlingS2[k, j]*(x)^(j)* D[Divide[1 - (x)^(n + 1),1 - x], {x, j}], {j, 0, k}] Failure Failure Skip Successful
26.8.E35 βˆ‘ j = 0 n j k = βˆ‘ j = 0 k j ! ⁒ S ⁑ ( k , j ) ⁒ ( n + 1 j + 1 ) superscript subscript 𝑗 0 𝑛 superscript 𝑗 π‘˜ superscript subscript 𝑗 0 π‘˜ 𝑗 Stirling-number-second-kind-S π‘˜ 𝑗 binomial 𝑛 1 𝑗 1 {\displaystyle{\displaystyle\sum_{j=0}^{n}j^{k}=\sum_{j=0}^{k}j!S\left(k,j% \right)\genfrac{(}{)}{0.0pt}{}{n+1}{j+1}}} sum((j)^(k), j = 0..n)= sum(factorial(j)*Stirling2(k, j)*binomial(n + 1,j + 1), j = 0..k) Sum[(j)^(k), {j, 0, n}]= Sum[(j)!*StirlingS2[k, j]*Binomial[n + 1,j + 1], {j, 0, k}] Failure Failure Skip Successful
26.8.E36 βˆ‘ k = 0 n ( - 1 ) n - k ⁒ k ! ⁒ S ⁑ ( n , k ) = 1 superscript subscript π‘˜ 0 𝑛 superscript 1 𝑛 π‘˜ π‘˜ Stirling-number-second-kind-S 𝑛 π‘˜ 1 {\displaystyle{\displaystyle\sum_{k=0}^{n}(-1)^{n-k}k!S\left(n,k\right)=1}} sum((- 1)^(n - k)* factorial(k)*Stirling2(n, k), k = 0..n)= 1 Sum[(- 1)^(n - k)* (k)!*StirlingS2[n, k], {k, 0, n}]= 1 Failure Failure Skip Successful
26.8.E39 βˆ‘ j = k n s ⁑ ( j , k ) ⁒ S ⁑ ( n , j ) = βˆ‘ j = k n s ⁑ ( n , j ) ⁒ S ⁑ ( j , k ) superscript subscript 𝑗 π‘˜ 𝑛 Stirling-number-first-kind-S 𝑗 π‘˜ Stirling-number-second-kind-S 𝑛 𝑗 superscript subscript 𝑗 π‘˜ 𝑛 Stirling-number-first-kind-S 𝑛 𝑗 Stirling-number-second-kind-S 𝑗 π‘˜ {\displaystyle{\displaystyle\sum_{j=k}^{n}s\left(j,k\right)S\left(n,j\right)=% \sum_{j=k}^{n}s\left(n,j\right)S\left(j,k\right)}} sum(Stirling1(j, k)*Stirling2(n, j), j = k..n)= sum(Stirling1(n, j)*Stirling2(j, k), j = k..n) Sum[StirlingS1[j, k]*StirlingS2[n, j], {j, k, n}]= Sum[StirlingS1[n, j]*StirlingS2[j, k], {j, k, n}] Failure Failure Skip
Fail
Complex[0.0, -2.8284271247461903] <- {Rule[Sum[Times[StirlingS1[n, j], StirlingS2[j, k]], {j, k, n}], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Sum[Times[StirlingS1[j, k], StirlingS2[n, j]], {j, k, n}], Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[-2.8284271247461903, -2.8284271247461903] <- {Rule[Sum[Times[StirlingS1[n, j], StirlingS2[j, k]], {j, k, n}], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Sum[Times[StirlingS1[j, k], StirlingS2[n, j]], {j, k, n}], Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
-2.8284271247461903 <- {Rule[Sum[Times[StirlingS1[n, j], StirlingS2[j, k]], {j, k, n}], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Sum[Times[StirlingS1[j, k], StirlingS2[n, j]], {j, k, n}], Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
Complex[0.0, 2.8284271247461903] <- {Rule[Sum[Times[StirlingS1[n, j], StirlingS2[j, k]], {j, k, n}], Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[Sum[Times[StirlingS1[j, k], StirlingS2[n, j]], {j, k, n}], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
26.8.E39 βˆ‘ j = k n s ⁑ ( n , j ) ⁒ S ⁑ ( j , k ) = Ξ΄ n , k superscript subscript 𝑗 π‘˜ 𝑛 Stirling-number-first-kind-S 𝑛 𝑗 Stirling-number-second-kind-S 𝑗 π‘˜ Kronecker 𝑛 π‘˜ {\displaystyle{\displaystyle\sum_{j=k}^{n}s\left(n,j\right)S\left(j,k\right)=% \delta_{n,k}}} sum(Stirling1(n, j)*Stirling2(j, k), j = k..n)= KroneckerDelta[n, k] Sum[StirlingS1[n, j]*StirlingS2[j, k], {j, k, n}]= KroneckerDelta[n, k] Failure Failure Skip Successful
26.10.E2 ∏ j = 1 ∞ ( 1 + q j ) = ∏ j = 1 ∞ 1 1 - q 2 ⁒ j - 1 superscript subscript product 𝑗 1 1 superscript π‘ž 𝑗 superscript subscript product 𝑗 1 1 1 superscript π‘ž 2 𝑗 1 {\displaystyle{\displaystyle\prod_{j=1}^{\infty}(1+q^{j})=\prod_{j=1}^{\infty}% \frac{1}{1-q^{2j-1}}}} product(1 + (q)^(j), j = 1..infinity)= product((1)/(1 - (q)^(2*j - 1)), j = 1..infinity) Product[1 + (q)^(j), {j, 1, Infinity}]= Product[Divide[1,1 - (q)^(2*j - 1)], {j, 1, Infinity}] Failure Failure Skip Successful
26.12.E23 ∏ h = 1 r 1 - q 3 ⁒ h - 1 1 - q 3 ⁒ h - 2 ⁒ ∏ 1 ≀ h < j ≀ r 1 - q 3 ⁒ ( h + 2 ⁒ j - 1 ) 1 - q 3 ⁒ ( h + j - 1 ) = ∏ h = 1 r ( 1 - q 3 ⁒ h - 1 1 - q 3 ⁒ h - 2 ⁒ ∏ j = h r 1 - q 3 ⁒ ( r + h + j - 1 ) 1 - q 3 ⁒ ( 2 ⁒ h + j - 1 ) ) superscript subscript product β„Ž 1 π‘Ÿ 1 superscript π‘ž 3 β„Ž 1 1 superscript π‘ž 3 β„Ž 2 subscript product 1 β„Ž 𝑗 π‘Ÿ 1 superscript π‘ž 3 β„Ž 2 𝑗 1 1 superscript π‘ž 3 β„Ž 𝑗 1 superscript subscript product β„Ž 1 π‘Ÿ 1 superscript π‘ž 3 β„Ž 1 1 superscript π‘ž 3 β„Ž 2 superscript subscript product 𝑗 β„Ž π‘Ÿ 1 superscript π‘ž 3 π‘Ÿ β„Ž 𝑗 1 1 superscript π‘ž 3 2 β„Ž 𝑗 1 {\displaystyle{\displaystyle\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)}} 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) Product[Divide[1 - (q)^(3*h - 1),1 - (q)^(3*h - 2)], {h, 1, r}]*Product[Product[Divide[1 - (q)^(3*(h + 2*j - 1)),1 - (q)^(3*(h + j - 1))], {j, h + 1, r}], {h, 1, j - 1}]= 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}], {h, 1, r}] Failure Failure Skip Error
26.12#Ex7 ΞΆ ⁑ ( 3 ) = 1.20205 69032 Riemann-zeta 3 1.20205 69032 {\displaystyle{\displaystyle\zeta\left(3\right)=1.20205\;69032}} Zeta(3)= 1.2020569032 Zeta[3]= 1.2020569032 Successful Failure - Successful
26.12#Ex8 ΞΆ β€² ⁑ ( - 1 ) = - 0.16542 11437 diffop Riemann-zeta 1 1 0.16542 11437 {\displaystyle{\displaystyle\zeta'\left(-1\right)=-0.16542\;11437}} subs( temp=- 1, diff( Zeta(temp), temp$(1) ) )= - 0.1654211437 (D[Zeta[temp], {temp, 1}]/.temp-> - 1)= - 0.1654211437 Successful Failure - Successful
26.13.E4 d ⁒ ( n ) = n ! ⁒ βˆ‘ j = 0 n ( - 1 ) j ⁒ 1 j ! 𝑑 𝑛 𝑛 superscript subscript 𝑗 0 𝑛 superscript 1 𝑗 1 𝑗 {\displaystyle{\displaystyle d(n)=n!\sum_{j=0}^{n}(-1)^{j}\frac{1}{j!}}} d*(n)= factorial(n)*sum((- 1)^(j)*(1)/(factorial(j)), j = 0..n) d*(n)= (n)!*Sum[(- 1)^(j)*Divide[1,(j)!], {j, 0, n}] Failure Failure Skip
Fail
Complex[1.4142135623730951, 1.4142135623730951] <- {Rule[d, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[n, 1]}
Complex[1.8284271247461903, 2.8284271247461903] <- {Rule[d, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[n, 2]}
Complex[2.2426406871192857, 4.242640687119286] <- {Rule[d, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[n, 3]}
Complex[1.4142135623730951, -1.4142135623730951] <- {Rule[d, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[n, 1]}
... skip entries to safe data
26.13.E4 n ! ⁒ βˆ‘ j = 0 n ( - 1 ) j ⁒ 1 j ! = ⌊ n ! + e - 2 e βŒ‹ 𝑛 superscript subscript 𝑗 0 𝑛 superscript 1 𝑗 1 𝑗 𝑛 2 {\displaystyle{\displaystyle n!\sum_{j=0}^{n}(-1)^{j}\frac{1}{j!}=\left\lfloor% \frac{n!+\mathrm{e}-2}{\mathrm{e}}\right\rfloor}} factorial(n)*sum((- 1)^(j)*(1)/(factorial(j)), j = 0..n)= floor((factorial(n)+ exp(1)- 2)/(exp(1))) (n)!*Sum[(- 1)^(j)*Divide[1,(j)!], {j, 0, n}]= Floor[Divide[(n)!+ E - 2,E]] Failure Failure Skip Successful
26.15.E9 r k ⁒ ( B ) = 2 ⁒ n 2 ⁒ n - k ⁒ ( 2 ⁒ n - k k ) subscript π‘Ÿ π‘˜ 𝐡 2 𝑛 2 𝑛 π‘˜ binomial 2 𝑛 π‘˜ π‘˜ {\displaystyle{\displaystyle r_{k}(B)=\frac{2n}{2n-k}\genfrac{(}{)}{0.0pt}{}{2% n-k}{k}}} r[k]*(B)=(2*n)/(2*n - k)*binomial(2*n - k,k) Subscript[r, k]*(B)=Divide[2*n,2*n - k]*Binomial[2*n - k,k] Failure Failure
Fail
-2.000000000+3.999999998*I <- {B = 2^(1/2)+I*2^(1/2), r[k] = 2^(1/2)+I*2^(1/2), k = 1, n = 1}
-4.000000000+3.999999998*I <- {B = 2^(1/2)+I*2^(1/2), r[k] = 2^(1/2)+I*2^(1/2), k = 1, n = 2}
-6.000000000+3.999999998*I <- {B = 2^(1/2)+I*2^(1/2), r[k] = 2^(1/2)+I*2^(1/2), k = 1, n = 3}
Float(-infinity)+3.999999998*I <- {B = 2^(1/2)+I*2^(1/2), r[k] = 2^(1/2)+I*2^(1/2), k = 2, n = 1}
... skip entries to safe data
 Successful
26.15.E10 2 ⁒ ( n ! ) ⁒ N 0 ⁒ ( B ) = 2 ⁒ ( n ! ) ⁒ βˆ‘ k = 0 n ( - 1 ) k ⁒ 2 ⁒ n 2 ⁒ n - k ⁒ ( 2 ⁒ n - k k ) ⁒ ( n - k ) ! 2 𝑛 subscript 𝑁 0 𝐡 2 𝑛 superscript subscript π‘˜ 0 𝑛 superscript 1 π‘˜ 2 𝑛 2 𝑛 π‘˜ binomial 2 𝑛 π‘˜ π‘˜ 𝑛 π‘˜ {\displaystyle{\displaystyle 2(n!)N_{0}(B)=2(n!)\sum_{k=0}^{n}(-1)^{k}\frac{2n% }{2n-k}\genfrac{(}{)}{0.0pt}{}{2n-k}{k}{(n-k)!}}} 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) 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}] Failure Failure Skip Successful
26.15.E13 r n - k ⁒ ( B ) = S ⁑ ( n , k ) subscript π‘Ÿ 𝑛 π‘˜ 𝐡 Stirling-number-second-kind-S 𝑛 π‘˜ {\displaystyle{\displaystyle r_{n-k}(B)=S\left(n,k\right)}} r[n - k]*(B)= Stirling2(n, k) Subscript[r, n - k]*(B)= StirlingS2[n, k] Failure Failure
Fail
-1.+3.999999998*I <- {B = 2^(1/2)+I*2^(1/2), r[n-k] = 2^(1/2)+I*2^(1/2), k = 1, n = 1}
-1.+3.999999998*I <- {B = 2^(1/2)+I*2^(1/2), r[n-k] = 2^(1/2)+I*2^(1/2), k = 1, n = 2}
-1.+3.999999998*I <- {B = 2^(1/2)+I*2^(1/2), r[n-k] = 2^(1/2)+I*2^(1/2), k = 1, n = 3}
0.+3.999999998*I <- {B = 2^(1/2)+I*2^(1/2), r[n-k] = 2^(1/2)+I*2^(1/2), k = 2, n = 1}
... skip entries to safe data
Fail
Complex[-1.4142135623730951, 2.585786437626905] <- {Rule[B, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[StirlingS2[n, k], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Subscript[r, Plus[Times[-1, k], n]], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[2.585786437626905, -1.4142135623730951] <- {Rule[B, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[StirlingS2[n, k], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Subscript[r, Plus[Times[-1, k], n]], Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[-1.4142135623730951, -5.414213562373095] <- {Rule[B, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[StirlingS2[n, k], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Subscript[r, Plus[Times[-1, k], n]], Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[-5.414213562373095, -1.4142135623730951] <- {Rule[B, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[StirlingS2[n, k], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Subscript[r, Plus[Times[-1, k], n]], Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
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