Combinatorial Analysis - 26.7 Set Partitions: Bell Numbers

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DLMF Formula Constraints Maple Mathematica Symbolic
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Mathematica 		Numeric
Maple 		Numeric
Mathematica
26.7.E1 B ⁡ ( 0 ) = 1 Bell-number 0 1 {\displaystyle{\displaystyle B\left(0\right)=1}}
\Bellnumber@{0} = 1		 

BellB(0, 1) = 1

BellB[0] == 1
Successful Successful - Successful [Tested: 1]
26.7.E2 B ⁡ ( n ) = ∑ k = 0 n S ⁡ ( n , k ) Bell-number 𝑛 superscript subscript 𝑘 0 𝑛 Stirling-number-second-kind-S 𝑛 𝑘 {\displaystyle{\displaystyle B\left(n\right)=\sum_{k=0}^{n}S\left(n,k\right)}}
\Bellnumber@{n} = \sum_{k=0}^{n}\StirlingnumberS@{n}{k}		 

BellB(n, 1) = sum(Stirling2(n, k), k = 0..n)

BellB[n] == Sum[StirlingS2[n, k], {k, 0, n}, GenerateConditions->None]
Failure Successful Successful [Tested: 3] Successful [Tested: 3]
26.7.E3 B ⁡ ( n ) = ∑ k = 1 m k n k ! ⁢ ∑ j = 0 m - k ( - 1 ) j j ! Bell-number 𝑛 superscript subscript 𝑘 1 𝑚 superscript 𝑘 𝑛 𝑘 superscript subscript 𝑗 0 𝑚 𝑘 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!}}}
\Bellnumber@{n} = \sum_{k=1}^{m}\frac{k^{n}}{k!}\sum_{j=0}^{m-k}\frac{(-1)^{j}}{j!}		 m ≥ n 𝑚 𝑛 {\displaystyle{\displaystyle m\geq n}} 
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}, GenerateConditions->None], {k, 1, m}, GenerateConditions->None]
Error Failure - Successful [Tested: 6]
26.7.E4 B ⁡ ( n ) = e - 1 ⁢ ∑ k = 1 ∞ k n k ! Bell-number 𝑛 1 superscript subscript 𝑘 1 superscript 𝑘 𝑛 𝑘 {\displaystyle{\displaystyle B\left(n\right)={\mathrm{e}^{-1}}\sum_{k=1}^{% \infty}\frac{k^{n}}{k!}}}
\Bellnumber@{n} = \expe^{-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}, GenerateConditions->None]
Failure Failure Successful [Tested: 3] Successful [Tested: 3]
26.7.E4 e - 1 ⁢ ∑ k = 1 ∞ k n k ! = 1 + ⌊ e - 1 ⁢ ∑ k = 1 2 ⁢ n k n k ! ⌋ 1 superscript subscript 𝑘 1 superscript 𝑘 𝑛 𝑘 1 1 superscript subscript 𝑘 1 2 𝑛 superscript 𝑘 𝑛 𝑘 {\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}}
\expe^{-1}\sum_{k=1}^{\infty}\frac{k^{n}}{k!} = 1+\floor{\expe^{-1}\sum_{k=1}^{2n}\frac{k^{n}}{k!}}		 

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}, GenerateConditions->None] == 1 + Floor[Exp[- 1]*Sum[Divide[(k)^(n),(k)!], {k, 1, 2*n}, GenerateConditions->None]]
Failure Failure Successful [Tested: 3] Successful [Tested: 3]
26.7.E5 ∑ n = 0 ∞ B ⁡ ( n ) ⁢ x n n ! = exp ⁡ ( e x - 1 ) superscript subscript 𝑛 0 Bell-number 𝑛 superscript 𝑥 𝑛 𝑛 𝑥 1 {\displaystyle{\displaystyle\sum_{n=0}^{\infty}B\left(n\right)\frac{x^{n}}{n!}% =\exp({\mathrm{e}^{x}}-1)}}
\sum_{n=0}^{\infty}\Bellnumber@{n}\frac{x^{n}}{n!} = \exp(\expe^{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}, GenerateConditions->None] == Exp[Exp[x]- 1]
Translation Error Translation Error - -
26.7.E6 B ⁡ ( n + 1 ) = ∑ k = 0 n ( n k ) ⁢ B ⁡ ( k ) Bell-number 𝑛 1 superscript subscript 𝑘 0 𝑛 binomial 𝑛 𝑘 Bell-number 𝑘 {\displaystyle{\displaystyle B\left(n+1\right)=\sum_{k=0}^{n}\genfrac{(}{)}{0.% 0pt}{}{n}{k}B\left(k\right)}}
\Bellnumber@{n+1} = \sum_{k=0}^{n}\binom{n}{k}\Bellnumber@{k}		 

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}, GenerateConditions->None]
Failure Failure Successful [Tested: 3] Successful [Tested: 3]
26.7#Ex1 B ⁡ ( n + 1 ) = ∑ k = 0 n ( n k ) ⁢ B ⁡ ( n ) Bell-number 𝑛 1 superscript subscript 𝑘 0 𝑛 binomial 𝑛 𝑘 Bell-number 𝑛 {\displaystyle{\displaystyle B\left(n+1\right)=\sum_{k=0}^{n}\genfrac{(}{)}{0.% 0pt}{}{n}{k}B\left(n\right)}}
\Bellnumber@{n+1} = \sum_{k=0}^{n}\binom{n}{k}\Bellnumber@{n}		 

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}, GenerateConditions->None]
Failure Failure
Failed [2 / 3]
Result: -3.
Test Values: {n = 2}


Result: -25.
Test Values: {n = 3}

Failed [2 / 3]
Result: -3.0
Test Values: {Rule[n, 2]}


Result: -25.0
Test Values: {Rule[n, 3]}

26.7.E8 N ⁢ ln ⁡ N = n 𝑁 𝑁 𝑛 {\displaystyle{\displaystyle N\ln N=n}}
N\ln@@{N} = n		 

N*ln(N) = n

N*Log[N] == n
Failure Failure
Failed [30 / 30]
Result: -1.261799388+.4534498412*I
Test Values: {N = 1/2*3^(1/2)+1/2*I, n = 1}


Result: -2.261799388+.4534498412*I
Test Values: {N = 1/2*3^(1/2)+1/2*I, n = 2}

... skip entries to safe data
Failed [30 / 30]
Result: Complex[-1.2617993877991494, 0.4534498410585544]
Test Values: {Rule[n, 1], Rule[N, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-2.261799387799149, 0.4534498410585544]
Test Values: {Rule[n, 2], Rule[N, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
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