A122418 a(n) = Sum_{k=0..n} (k-1)^n*k!*Stirling2(n,k).
1, 0, 2, 54, 2534, 186030, 19794662, 2885980734, 552803552534, 134687987183790, 40686498089484422, 14925683377452413214, 6536580413039406774134, 3368723388994026165415950, 2018248855531992511720945382, 1390953089533285777007059354494, 1092714503596231472933813958469334
Offset: 0
Links
- G. C. Greubel, Table of n, a(n) for n = 0..229
Programs
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Maple
A122418 := proc(n) sum((k-1)^n*k!*combinat[stirling2](n,k),k=0..n) ; end; for n from 0 to 16 do print(A122418(n)) ; od ; # R. J. Mathar, Feb 10 2007
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Mathematica
a[n_] := Sum[ (k-1)^n*k!*StirlingS2[n, k], {k, 0, n}]; Table[a[n], {n, 0, 16}] (* Jean-François Alcover, Mar 26 2013 *) -
PARI
for(n=0,50, print1(sum(k=0,n, (k-1)^n*k!*stirling(n,k,2)), ", ")) \\ G. C. Greubel, Nov 15 2017
Formula
E.g.f.: Sum((exp((n-1)*x)-1)^n, n=0..infinity).
a(n) ~ c * d^n * (n!)^2 / sqrt(n), where d = A317855 = (1+exp(1/r))*r^2 = 3.161088653865428813830172202588132491726382774188556341627278..., r = 0.8737024332396683304965683047207192982139922672025395099... is the root of the equation exp(1/r)/r + (1+exp(1/r))*LambertW(-exp(-1/r)/r) = 0, and c = 0.10430562057820038909699083625848223918044424242153125547162600916636313858475... . - Vaclav Kotesovec, May 07 2014
Extensions
More terms from R. J. Mathar, Feb 10 2007