A122419 Number of labeled digraphs with n arcs and with no vertex of indegree 0.
1, 0, 1, 8, 93, 1354, 23900, 496244, 11855700, 320428318, 9667220397, 322072882348, 11744421711587, 465270864839688, 19899234175413257, 913836170567749048, 44849438199960187278, 2342666125012348876152
Offset: 0
Programs
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Maple
A122418 := proc(n) option remember ; add( combinat[stirling2](n,k)*(k-1)^n*k!,k=0..n) ; end: A122419 := proc(n) option remember ; add( combinat[stirling1](n,k)*A122418(k),k=0..n)/n! ; end: for n from 0 to 30 do printf("%d, ",A122419(n)) ; od ; # R. J. Mathar, May 18 2007
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Mathematica
nmax=20; CoefficientList[Series[Sum[((1+x)^(n-1)-1)^n, {n, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, May 06 2014 *)
Formula
a(n) = (1/n!)*Sum_{k=0..n} Stirling1(n,k)*A122418(k).
G.f.: Sum_{n>=0} ((1+x)^(n-1) - 1)^n.
a(n) ~ c * d^n * n! / 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 = 0.08904589343883135100956914504938... . - Vaclav Kotesovec, May 07 2014
Extensions
More terms from R. J. Mathar, May 18 2007