Combinatorial Analysis - 26.14 Permutations: Order Notation

From testwiki
Revision as of 12:06, 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 → (diff)
Jump to navigation Jump to search


DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
26.14.E4 ∑ n , k = 0 ∞ ⟨ n k ⟩ ⁢ x k ⁢ t n n ! = 1 - x exp ⁡ ( ( x - 1 ) ⁢ t ) - x superscript subscript 𝑛 𝑘 0 Eulerian-number 𝑛 𝑘 superscript 𝑥 𝑘 superscript 𝑡 𝑛 𝑛 1 𝑥 𝑥 1 𝑡 𝑥 {\displaystyle{\displaystyle\sum_{n,k=0}^{\infty}\genfrac{<}{>}{0.0pt}{}{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}		 | x | < 1 , | t | < 1 formulae-sequence 𝑥 1 𝑡 1 {\displaystyle{\displaystyle|x|<1,|t|<1}} 
Error

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]
Missing Macro Error Translation Error - -
26.14.E5 ∑ k = 0 n - 1 ⟨ n k ⟩ ⁢ ( x + k n ) = x n superscript subscript 𝑘 0 𝑛 1 Eulerian-number 𝑛 𝑘 binomial 𝑥 𝑘 𝑛 superscript 𝑥 𝑛 {\displaystyle{\displaystyle\sum_{k=0}^{n-1}\genfrac{<}{>}{0.0pt}{}{n}{k}% \genfrac{(}{)}{0.0pt}{}{x+k}{n}=x^{n}}}
\sum_{k=0}^{n-1}\Euleriannumber{n}{k}\binom{x+k}{n} = x^{n}		 

Error

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)
Missing Macro Error Failure - Successful [Tested: 9]
26.14.E6 ⟨ n k ⟩ = ∑ j = 0 k ( - 1 ) j ⁢ ( n + 1 j ) ⁢ ( k + 1 - j ) n Eulerian-number 𝑛 𝑘 superscript subscript 𝑗 0 𝑘 superscript 1 𝑗 binomial 𝑛 1 𝑗 superscript 𝑘 1 𝑗 𝑛 {\displaystyle{\displaystyle\genfrac{<}{>}{0.0pt}{}{n}{k}=\sum_{j=0}^{k}(-1)^{% j}\genfrac{(}{)}{0.0pt}{}{n+1}{j}(k+1-j)^{n}}}
\Euleriannumber{n}{k} = \sum_{j=0}^{k}(-1)^{j}\binom{n+1}{j}(k+1-j)^{n}		 n ≥ 1 𝑛 1 {\displaystyle{\displaystyle n\geq 1}} 
Error

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]
Missing Macro Error Failure - Successful [Tested: 9]
26.14.E7 ⟨ n k ⟩ = ∑ j = 0 n - k ( - 1 ) n - k - j ⁢ j ! ⁢ ( n - j k ) ⁢ S ⁡ ( n , j ) Eulerian-number 𝑛 𝑘 superscript subscript 𝑗 0 𝑛 𝑘 superscript 1 𝑛 𝑘 𝑗 𝑗 binomial 𝑛 𝑗 𝑘 Stirling-number-second-kind-S 𝑛 𝑗 {\displaystyle{\displaystyle\genfrac{<}{>}{0.0pt}{}{n}{k}=\sum_{j=0}^{n-k}(-1)% ^{n-k-j}j!\genfrac{(}{)}{0.0pt}{}{n-j}{k}S\left(n,j\right)}}
\Euleriannumber{n}{k} = \sum_{j=0}^{n-k}(-1)^{n-k-j}j!\binom{n-j}{k}\StirlingnumberS@{n}{j}		 

Error

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]
Missing Macro Error Failure - Successful [Tested: 9]
26.14.E8 ⟨ n k ⟩ = ( k + 1 ) ⁢ ⟨ n - 1 k ⟩ + ( n - k ) ⁢ ⟨ n - 1 k - 1 ⟩ Eulerian-number 𝑛 𝑘 𝑘 1 Eulerian-number 𝑛 1 𝑘 𝑛 𝑘 Eulerian-number 𝑛 1 𝑘 1 {\displaystyle{\displaystyle\genfrac{<}{>}{0.0pt}{}{n}{k}=(k+1)\genfrac{<}{>}{% 0.0pt}{}{n-1}{k}+(n-k)\genfrac{<}{>}{0.0pt}{}{n-1}{k-1}}}
\Euleriannumber{n}{k} = (k+1)\Euleriannumber{n-1}{k}+(n-k)\Euleriannumber{n-1}{k-1}		 n ≥ 2 𝑛 2 {\displaystyle{\displaystyle n\geq 2}} 
Error

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}]
Missing Macro Error Failure - Successful [Tested: 6]
26.14.E9 ⟨ n k ⟩ = ⟨ n n - 1 - k ⟩ Eulerian-number 𝑛 𝑘 Eulerian-number 𝑛 𝑛 1 𝑘 {\displaystyle{\displaystyle\genfrac{<}{>}{0.0pt}{}{n}{k}=\genfrac{<}{>}{0.0pt% }{}{n}{n-1-k}}}
\Euleriannumber{n}{k} = \Euleriannumber{n}{n-1-k}		 n ≥ 1 𝑛 1 {\displaystyle{\displaystyle n\geq 1}} 
Error

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}]
Missing Macro Error Failure - Successful [Tested: 9]
26.14.E10 ∑ k = 0 n - 1 ⟨ n k ⟩ = n ! superscript subscript 𝑘 0 𝑛 1 Eulerian-number 𝑛 𝑘 𝑛 {\displaystyle{\displaystyle\sum_{k=0}^{n-1}\genfrac{<}{>}{0.0pt}{}{n}{k}=n!}}
\sum_{k=0}^{n-1}\Euleriannumber{n}{k} = n!		 n ≥ 1 𝑛 1 {\displaystyle{\displaystyle n\geq 1}} 
Error

Sum[Sum[(-1)^m Binomial[n+1,m] (k-m+1)^(n),{m,0,k+1}], {k, 0, n - 1}, GenerateConditions->None] == (n)!
Missing Macro Error Failure - Successful [Tested: 3]
26.14.E11 B m = m 2 m ⁢ ( 2 m - 1 ) ⁢ ∑ k = 0 m - 2 ( - 1 ) k ⁢ ⟨ m - 1 k ⟩ Bernoulli-number-B 𝑚 𝑚 superscript 2 𝑚 superscript 2 𝑚 1 superscript subscript 𝑘 0 𝑚 2 superscript 1 𝑘 Eulerian-number 𝑚 1 𝑘 {\displaystyle{\displaystyle B_{m}=\frac{m}{2^{m}(2^{m}-1)}\sum_{k=0}^{m-2}(-1% )^{k}\genfrac{<}{>}{0.0pt}{}{m-1}{k}}}
\BernoullinumberB{m} = \frac{m}{2^{m}(2^{m}-1)}\sum_{k=0}^{m-2}(-1)^{k}\Euleriannumber{m-1}{k}		 m ≥ 2 𝑚 2 {\displaystyle{\displaystyle m\geq 2}} 
Error

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]
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]}

26.14.E12 S ⁡ ( n , m ) = 1 m ! ⁢ ∑ k = 0 n - 1 ⟨ n k ⟩ ⁢ ( k n - m ) Stirling-number-second-kind-S 𝑛 𝑚 1 𝑚 superscript subscript 𝑘 0 𝑛 1 Eulerian-number 𝑛 𝑘 binomial 𝑘 𝑛 𝑚 {\displaystyle{\displaystyle S\left(n,m\right)=\frac{1}{m!}\sum_{k=0}^{n-1}% \genfrac{<}{>}{0.0pt}{}{n}{k}\genfrac{(}{)}{0.0pt}{}{k}{n-m}}}
\StirlingnumberS@{n}{m} = \frac{1}{m!}\sum_{k=0}^{n-1}\Euleriannumber{n}{k}\binom{k}{n-m}		 n ≥ m , n ≥ 1 formulae-sequence 𝑛 𝑚 𝑛 1 {\displaystyle{\displaystyle n\geq m,n\geq 1}} 
Error

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]
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 ⟨ 0 k ⟩ = δ 0 , k Eulerian-number 0 𝑘 Kronecker 0 𝑘 {\displaystyle{\displaystyle\genfrac{<}{>}{0.0pt}{}{0}{k}=\delta_{0,k}}}
\Euleriannumber{0}{k} = \Kroneckerdelta{0}{k}		 

Error

Sum[(-1)^m Binomial[0+1,m] (k-m+1)^(0),{m,0,k+1}] == KroneckerDelta[0, k]
Missing Macro Error Failure - Successful [Tested: 3]
26.14.E14 ⟨ n 0 ⟩ = 1 Eulerian-number 𝑛 0 1 {\displaystyle{\displaystyle\genfrac{<}{>}{0.0pt}{}{n}{0}=1}}
\Euleriannumber{n}{0} = 1		 

Error

Sum[(-1)^m Binomial[n+1,m] (0-m+1)^(n),{m,0,0+1}] == 1
Missing Macro Error Failure - Successful [Tested: 3]
26.14.E15 ⟨ n 1 ⟩ = 2 n - n - 1 Eulerian-number 𝑛 1 superscript 2 𝑛 𝑛 1 {\displaystyle{\displaystyle\genfrac{<}{>}{0.0pt}{}{n}{1}=2^{n}-n-1}}
\Euleriannumber{n}{1} = 2^{n}-n-1		 n ≥ 1 𝑛 1 {\displaystyle{\displaystyle n\geq 1}} 
Error

Sum[(-1)^m Binomial[n+1,m] (1-m+1)^(n),{m,0,1+1}] == (2)^(n)- n - 1
Missing Macro Error Successful - Successful [Tested: 3]
26.14.E16 ⟨ n 2 ⟩ = 3 n - ( n + 1 ) ⁢ 2 n + ( n + 1 2 ) Eulerian-number 𝑛 2 superscript 3 𝑛 𝑛 1 superscript 2 𝑛 binomial 𝑛 1 2 {\displaystyle{\displaystyle\genfrac{<}{>}{0.0pt}{}{n}{2}=3^{n}-(n+1)2^{n}+% \genfrac{(}{)}{0.0pt}{}{n+1}{2}}}
\Euleriannumber{n}{2} = 3^{n}-(n+1)2^{n}+\binom{n+1}{2}		 n ≥ 1 𝑛 1 {\displaystyle{\displaystyle n\geq 1}} 
Error

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]
Missing Macro Error Successful - Successful [Tested: 3]
Retrieved from "https://lct.wmflabs.org/w/index.php?title=26.14&oldid=58469"