Combinatorial Analysis - 26.8 Set Partitions: Stirling Numbers

From testwiki
Revision as of 12:05, 28 June 2021 by Admin (talk | contribs) (Admin moved page Main Page to Verifying DLMF with Maple and Mathematica)
(diff) ← Older revision | Latest revision (diff) | Newer revision β†’ (diff)
Jump to navigation Jump to search


DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
26.8.E1 s ⁑ ( n , n ) = 1 Stirling-number-first-kind-S 𝑛 𝑛 1 {\displaystyle{\displaystyle s\left(n,n\right)=1}}
\Stirlingnumbers@{n}{n} = 1		 n β‰₯ 0 𝑛 0 {\displaystyle{\displaystyle n\geq 0}} 
Stirling1(n, n) = 1

StirlingS1[n, n] == 1
Successful Failure - Successful [Tested: 3]
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}}}
\Stirlingnumbers@{1}{k} = \Kroneckerdelta{1}{k}		 

Stirling1(1, k) = KroneckerDelta[1, k]

StirlingS1[1, k] == KroneckerDelta[1, k]
Failure Failure Successful [Tested: 3] Successful [Tested: 3]
26.8.E4 S ⁑ ( n , n ) = 1 Stirling-number-second-kind-S 𝑛 𝑛 1 {\displaystyle{\displaystyle S\left(n,n\right)=1}}
\StirlingnumberS@{n}{n} = 1		 n β‰₯ 0 𝑛 0 {\displaystyle{\displaystyle n\geq 0}} 
Stirling2(n, n) = 1

StirlingS2[n, n] == 1
Successful Failure - Successful [Tested: 3]
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}}}
\StirlingnumberS@{n}{k} = \frac{1}{k!}\sum_{j=0}^{k}(-1)^{k-j}\binom{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}, GenerateConditions->None]
Aborted Failure Successful [Tested: 9] Successful [Tested: 9]
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_{k=0}^{n}\Stirlingnumbers@{n}{k}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}, GenerateConditions->None] == Subscript[x - n + 1, n]
Failure Failure Error
Failed [9 / 9]
Result: Plus[1.5, Times[-1.0, Subscript[1.5, 1]]]
Test Values: {Rule[n, 1], Rule[x, 1.5]}


Result: Plus[0.75, Times[-1.0, Subscript[0.5, 2]]]
Test Values: {Rule[n, 2], Rule[x, 1.5]}

... skip entries to safe data
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_{n=0}^{\infty}\Stirlingnumbers@{n}{k}\frac{x^{n}}{n!} = \frac{(\ln@{1+x})^{k}}{k!}		 | x | < 1 π‘₯ 1 {\displaystyle{\displaystyle|x|<1}} 
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}, GenerateConditions->None] == Divide[(Log[1 + x])^(k),(k)!]
Error Failure -
Failed [2 / 3]
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]}


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]}

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_{n,k=0}^{\infty}\Stirlingnumbers@{n}{k}\frac{x^{n}}{n!}y^{k} = (1+x)^{y}		 | x | < 1 π‘₯ 1 {\displaystyle{\displaystyle|x|<1}} 
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}, GenerateConditions->None], {n, 0, Infinity}, GenerateConditions->None] == (1 + x)^(y)
Error Failure -
Failed [6 / 6]
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]}


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]}

... skip entries to safe data
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_{k=1}^{n}\StirlingnumberS@{n}{k}(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}, GenerateConditions->None] == (x)^(n)
Failure Failure Error
Failed [9 / 9]
Result: Plus[-1.5, Subscript[1.5, 1]]
Test Values: {Rule[n, 1], Rule[x, 1.5]}


Result: Plus[-2.25, Subscript[0.5, 2], Subscript[1.5, 1]]
Test Values: {Rule[n, 2], Rule[x, 1.5]}

... skip entries to safe data
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_{n=0}^{\infty}\StirlingnumberS@{n}{k}\frac{x^{n}}{n!} = \frac{(\expe^{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}, GenerateConditions->None] == Divide[(Exp[x]- 1)^(k),(k)!]
Failure Failure Error Successful [Tested: 9]
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_{n,k=0}^{\infty}\StirlingnumberS@{n}{k}\frac{x^{n}}{n!}y^{k} = \exp\left(y(\expe^{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}, GenerateConditions->None], {n, 0, Infinity}, GenerateConditions->None] == Exp[y*(Exp[x]- 1)]
Translation Error Translation Error - -
26.8#Ex1 s ⁑ ( n , 0 ) = 0 Stirling-number-first-kind-S 𝑛 0 0 {\displaystyle{\displaystyle s\left(n,0\right)=0}}
\Stirlingnumbers@{n}{0} = 0		 

Stirling1(n, 0) = 0

StirlingS1[n, 0] == 0
Failure Failure Successful [Tested: 3] Successful [Tested: 3]
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)!}}
\Stirlingnumbers@{n}{1} = (-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 [Tested: 3] Successful [Tested: 3]
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)}}
-\Stirlingnumbers@{n}{n-1} = \StirlingnumberS@{n}{n-1}		 

- Stirling1(n, n - 1) = Stirling2(n, n - 1)

- StirlingS1[n, n - 1] == StirlingS2[n, n - 1]
Successful Failure - Successful [Tested: 3]
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}}}
\StirlingnumberS@{n}{n-1} = \binom{n}{2}		 

Stirling2(n, n - 1) = binomial(n,2)

StirlingS2[n, n - 1] == Binomial[n,2]
Successful Failure - Successful [Tested: 3]
26.8#Ex3 S ⁑ ( n , 0 ) = 0 Stirling-number-second-kind-S 𝑛 0 0 {\displaystyle{\displaystyle S\left(n,0\right)=0}}
\StirlingnumberS@{n}{0} = 0		 

Stirling2(n, 0) = 0

StirlingS2[n, 0] == 0
Failure Failure Successful [Tested: 3] Successful [Tested: 3]
26.8#Ex4 S ⁑ ( n , 1 ) = 1 Stirling-number-second-kind-S 𝑛 1 1 {\displaystyle{\displaystyle S\left(n,1\right)=1}}
\StirlingnumberS@{n}{1} = 1		 

Stirling2(n, 1) = 1

StirlingS2[n, 1] == 1
Failure Failure Successful [Tested: 3] Successful [Tested: 3]
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}}
\StirlingnumberS@{n}{2} = 2^{n-1}-1		 

Stirling2(n, 2) = (2)^(n - 1)- 1

StirlingS2[n, 2] == (2)^(n - 1)- 1
Failure Failure Successful [Tested: 3] Successful [Tested: 3]
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)}}
\Stirlingnumbers@{n}{k} = \Stirlingnumbers@{n-1}{k-1}-(n-1)\Stirlingnumbers@{n-1}{k}		 

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 [Tested: 9] Successful [Tested: 9]
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)}}
\binom{k}{h}\Stirlingnumbers@{n}{k} = \sum_{j=k-h}^{n-h}\binom{n}{j}\Stirlingnumbers@{n-j}{h}\Stirlingnumbers@{j}{k-h}		 n β‰₯ k , k β‰₯ h formulae-sequence 𝑛 π‘˜ π‘˜ β„Ž {\displaystyle{\displaystyle n\geq k,k\geq h}} 
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}, GenerateConditions->None]
Error Failure -
Failed [11 / 30]
Result: 0.16976527263135505
Test Values: {Rule[h, -1.5], Rule[k, 1], Rule[n, 2]}


Result: -0.08488263631567752
Test Values: {Rule[h, -1.5], Rule[k, 1], Rule[n, 3]}

... skip entries to safe data
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)}}
\Stirlingnumbers@{n+1}{k+1} = n!\sum_{j=k}^{n}\frac{(-1)^{n-j}}{j!}\,\Stirlingnumbers@{j}{k}		 

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}, GenerateConditions->None]
Failure Failure Successful [Tested: 9] Successful [Tested: 9]
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)}}
\Stirlingnumbers@{n+k+1}{k} = -\sum_{j=0}^{k}(n+j)\Stirlingnumbers@{n+j}{j}		 

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}, GenerateConditions->None]
Failure Failure Successful [Tested: 9] Successful [Tested: 9]
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)}}
\StirlingnumberS@{n}{k} = k\StirlingnumberS@{n-1}{k}+\StirlingnumberS@{n-1}{k-1}		 

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 [Tested: 9] Successful [Tested: 9]
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)}}
\binom{k}{h}\StirlingnumberS@{n}{k} = \sum_{j=k-h}^{n-h}\binom{n}{j}\StirlingnumberS@{n-j}{h}\StirlingnumberS@{j}{k-h}		 n β‰₯ k , k β‰₯ h formulae-sequence 𝑛 π‘˜ π‘˜ β„Ž {\displaystyle{\displaystyle n\geq k,k\geq h}} 
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}, GenerateConditions->None]
Error Failure -
Failed [22 / 30]
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]}


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]}

... skip entries to safe data
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}}}
\StirlingnumberS@{n}{k} = \sum_{j=k}^{n}\StirlingnumberS@{j-1}{k-1}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}, GenerateConditions->None]
Failure Failure Successful [Tested: 9] Successful [Tested: 9]
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)}}
\StirlingnumberS@{n+1}{k+1} = \sum_{j=k}^{n}\binom{n}{j}\StirlingnumberS@{j}{k}		 

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}, GenerateConditions->None]
Failure Failure Successful [Tested: 9] Successful [Tested: 9]
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)}}
\StirlingnumberS@{n+k+1}{k} = \sum_{j=0}^{k}j\StirlingnumberS@{n+j}{j}		 

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}, GenerateConditions->None]
Failure Successful Successful [Tested: 9] Successful [Tested: 9]
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)}}
\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}		 

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}, GenerateConditions->None]
Failure Failure Successful [Tested: 9]
Failed [3 / 9]
Result: StirlingS1[1.0, -1.0]
Test Values: {Rule[k, 2], Rule[n, 1]}


Result: StirlingS1[1.0, -2.0]
Test Values: {Rule[k, 3], Rule[n, 1]}

... 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_{k=1}^{n}\Stirlingnumbers@{n}{k} = 0		 n > 1 𝑛 1 {\displaystyle{\displaystyle n>1}} 
sum(Stirling1(n, k), k = 1..n) = 0

Sum[StirlingS1[n, k], {k, 1, n}, GenerateConditions->None] == 0
Failure Failure Successful [Tested: 2] Successful [Tested: 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_{k=1}^{n}(-1)^{n-k}\Stirlingnumbers@{n}{k} = n!		 

sum((- 1)^(n - k)* Stirling1(n, k), k = 1..n) = factorial(n)

Sum[(- 1)^(n - k)* StirlingS1[n, k], {k, 1, n}, GenerateConditions->None] == (n)!
 Failure Failure Successful [Tested: 3] Successful [Tested: 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_{j=k}^{n}\Stirlingnumbers@{n+1}{j+1}\,n^{j-k} = \Stirlingnumbers@{n}{k}		 

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}, GenerateConditions->None] == StirlingS1[n, k]
 Failure Successful Successful [Tested: 9] Successful [Tested: 9]
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)}}
\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}		 

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}, GenerateConditions->None]
 Failure Failure Successful [Tested: 9]
Failed [3 / 9]
Result: StirlingS2[1.0, -1.0]
Test Values: {Rule[k, 2], Rule[n, 1]}

Result: StirlingS2[1.0, -2.0]
Test Values: {Rule[k, 3], Rule[n, 1]}

... 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=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)		 

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}, GenerateConditions->None] == Sum[StirlingS2[k, j]*(x)^(j)* D[Divide[1 - (x)^(n + 1),1 - x], {x, j}], {j, 0, k}, GenerateConditions->None]
 Aborted Failure Skipped - Because timed out Skipped - Because timed out
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=0}^{n}j^{k} = \sum_{j=0}^{k}j!\StirlingnumberS@{k}{j}\binom{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}, GenerateConditions->None] == Sum[(j)!*StirlingS2[k, j]*Binomial[n + 1,j + 1], {j, 0, k}, GenerateConditions->None]
 Failure Failure Successful [Tested: 9] Successful [Tested: 9]
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_{k=0}^{n}(-1)^{n-k}k!\StirlingnumberS@{n}{k} = 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}, GenerateConditions->None] == 1
 Failure Failure Successful [Tested: 3] Successful [Tested: 3]
26.8.E38 A - 1 = B superscript 𝐴 1 𝐡 {\displaystyle{\displaystyle A^{-1}=B}}
A^{-1} = B		 

(A)^(- 1) = B
 
(A)^(- 1) == B
 Skipped - no semantic math Skipped - no semantic math - -
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_{j=k}^{n}\Stirlingnumbers@{j}{k}\StirlingnumberS@{n}{j} = \sum_{j=k}^{n}\Stirlingnumbers@{n}{j}\StirlingnumberS@{j}{k}		 

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}, GenerateConditions->None] == Sum[StirlingS1[n, j]*StirlingS2[j, k], {j, k, n}, GenerateConditions->None]
 Failure Failure Successful [Tested: 9] Successful [Tested: 9]
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_{j=k}^{n}\Stirlingnumbers@{n}{j}\StirlingnumberS@{j}{k} = \Kroneckerdelta{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}, GenerateConditions->None] == KroneckerDelta[n, k]
 Failure Failure Successful [Tested: 9] Successful [Tested: 9]
Retrieved from "https://lct.wmflabs.org/w/index.php?title=26.8&oldid=58463"