Combinatorial Analysis - 26.7 Set Partitions: Bell Numbers
Jump to navigation
Jump to search
| DLMF | Formula | Constraints | Maple | Mathematica | Symbolic Maple |
Symbolic Mathematica |
Numeric Maple |
Numeric Mathematica |
|---|---|---|---|---|---|---|---|---|
| 26.7.E1 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Bellnumber@{0} = 1}
\Bellnumber@{0} = 1 |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | BellB(0, 1) = 1
|
BellB[0] == 1
|
Successful | Successful | - | Successful [Tested: 1] |
| 26.7.E2 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Bellnumber@{n} = \sum_{k=0}^{n}\StirlingnumberS@{n}{k}}
\Bellnumber@{n} = \sum_{k=0}^{n}\StirlingnumberS@{n}{k} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | 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 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Bellnumber@{n} = \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!} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\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 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Bellnumber@{n} = \expe^{-1}\sum_{k=1}^{\infty}\frac{k^{n}}{k!}}
\Bellnumber@{n} = \expe^{-1}\sum_{k=1}^{\infty}\frac{k^{n}}{k!} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | 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 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \expe^{-1}\sum_{k=1}^{\infty}\frac{k^{n}}{k!} = 1+\floor{\expe^{-1}\sum_{k=1}^{2n}\frac{k^{n}}{k!}}}
\expe^{-1}\sum_{k=1}^{\infty}\frac{k^{n}}{k!} = 1+\floor{\expe^{-1}\sum_{k=1}^{2n}\frac{k^{n}}{k!}} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | 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 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \sum_{n=0}^{\infty}\Bellnumber@{n}\frac{x^{n}}{n!} = \exp(\expe^{x}-1)}
\sum_{n=0}^{\infty}\Bellnumber@{n}\frac{x^{n}}{n!} = \exp(\expe^{x}-1) |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | 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 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Bellnumber@{n+1} = \sum_{k=0}^{n}\binom{n}{k}\Bellnumber@{k}}
\Bellnumber@{n+1} = \sum_{k=0}^{n}\binom{n}{k}\Bellnumber@{k} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | 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 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Bellnumber@{n+1} = \sum_{k=0}^{n}\binom{n}{k}\Bellnumber@{n}}
\Bellnumber@{n+1} = \sum_{k=0}^{n}\binom{n}{k}\Bellnumber@{n} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | 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 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle N\ln@@{N} = n}
N\ln@@{N} = n |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | 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 |