Combinatorial Analysis - 26.14 Permutations: Order Notation
| DLMF | Formula | Constraints | Maple | Mathematica | Symbolic Maple |
Symbolic Mathematica |
Numeric Maple |
Numeric Mathematica |
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| 26.14.E4 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \sum_{n,k=0}^{\infty}\Euleriannumber{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} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle |x| < 1, |t| < 1} | Error
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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]
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Missing Macro Error | Translation Error | - | - |
| 26.14.E5 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \sum_{k=0}^{n-1}\Euleriannumber{n}{k}\binom{x+k}{n} = x^{n}}
\sum_{k=0}^{n-1}\Euleriannumber{n}{k}\binom{x+k}{n} = x^{n} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | Error
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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)
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Missing Macro Error | Failure | - | Successful [Tested: 9] |
| 26.14.E6 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Euleriannumber{n}{k} = \sum_{j=0}^{k}(-1)^{j}\binom{n+1}{j}(k+1-j)^{n}}
\Euleriannumber{n}{k} = \sum_{j=0}^{k}(-1)^{j}\binom{n+1}{j}(k+1-j)^{n} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle n \geq 1} | Error
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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]
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Missing Macro Error | Failure | - | Successful [Tested: 9] |
| 26.14.E7 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Euleriannumber{n}{k} = \sum_{j=0}^{n-k}(-1)^{n-k-j}j!\binom{n-j}{k}\StirlingnumberS@{n}{j}}
\Euleriannumber{n}{k} = \sum_{j=0}^{n-k}(-1)^{n-k-j}j!\binom{n-j}{k}\StirlingnumberS@{n}{j} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | Error
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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]
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Missing Macro Error | Failure | - | Successful [Tested: 9] |
| 26.14.E8 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Euleriannumber{n}{k} = (k+1)\Euleriannumber{n-1}{k}+(n-k)\Euleriannumber{n-1}{k-1}}
\Euleriannumber{n}{k} = (k+1)\Euleriannumber{n-1}{k}+(n-k)\Euleriannumber{n-1}{k-1} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle n \geq 2} | Error
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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}]
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Missing Macro Error | Failure | - | Successful [Tested: 6] |
| 26.14.E9 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Euleriannumber{n}{k} = \Euleriannumber{n}{n-1-k}}
\Euleriannumber{n}{k} = \Euleriannumber{n}{n-1-k} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle n \geq 1} | Error
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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}]
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Missing Macro Error | Failure | - | Successful [Tested: 9] |
| 26.14.E10 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \sum_{k=0}^{n-1}\Euleriannumber{n}{k} = n!}
\sum_{k=0}^{n-1}\Euleriannumber{n}{k} = n! |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle n \geq 1} | Error
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Sum[Sum[(-1)^m Binomial[n+1,m] (k-m+1)^(n),{m,0,k+1}], {k, 0, n - 1}, GenerateConditions->None] == (n)!
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Missing Macro Error | Failure | - | Successful [Tested: 3] |
| 26.14.E11 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \BernoullinumberB{m} = \frac{m}{2^{m}(2^{m}-1)}\sum_{k=0}^{m-2}(-1)^{k}\Euleriannumber{m-1}{k}}
\BernoullinumberB{m} = \frac{m}{2^{m}(2^{m}-1)}\sum_{k=0}^{m-2}(-1)^{k}\Euleriannumber{m-1}{k} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle m \geq 2} | Error
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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]
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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]}
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| 26.14.E12 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \StirlingnumberS@{n}{m} = \frac{1}{m!}\sum_{k=0}^{n-1}\Euleriannumber{n}{k}\binom{k}{n-m}}
\StirlingnumberS@{n}{m} = \frac{1}{m!}\sum_{k=0}^{n-1}\Euleriannumber{n}{k}\binom{k}{n-m} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle n \geq m, n \geq 1} | Error
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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]
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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 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Euleriannumber{0}{k} = \Kroneckerdelta{0}{k}}
\Euleriannumber{0}{k} = \Kroneckerdelta{0}{k} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | Error
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Sum[(-1)^m Binomial[0+1,m] (k-m+1)^(0),{m,0,k+1}] == KroneckerDelta[0, k]
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Missing Macro Error | Failure | - | Successful [Tested: 3] |
| 26.14.E14 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Euleriannumber{n}{0} = 1}
\Euleriannumber{n}{0} = 1 |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | Error
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Sum[(-1)^m Binomial[n+1,m] (0-m+1)^(n),{m,0,0+1}] == 1
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Missing Macro Error | Failure | - | Successful [Tested: 3] |
| 26.14.E15 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Euleriannumber{n}{1} = 2^{n}-n-1}
\Euleriannumber{n}{1} = 2^{n}-n-1 |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle n \geq 1} | Error
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Sum[(-1)^m Binomial[n+1,m] (1-m+1)^(n),{m,0,1+1}] == (2)^(n)- n - 1
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Missing Macro Error | Successful | - | Successful [Tested: 3] |
| 26.14.E16 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Euleriannumber{n}{2} = 3^{n}-(n+1)2^{n}+\binom{n+1}{2}}
\Euleriannumber{n}{2} = 3^{n}-(n+1)2^{n}+\binom{n+1}{2} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle n \geq 1} | Error
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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]
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Missing Macro Error | Successful | - | Successful [Tested: 3] |