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

0,4

Links

Table of n, a(n) for n=0..17.

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

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

Mathematica

nmax=20; CoefficientList[Series[Sum[((1+x)^(n-1)-1)^n, {n, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, May 06 2014 *)

Crossrefs

Cf. A122420, A122400.

Sequence in context: A099291 A087579 A194043 * A319176 A299034 A299339

Adjacent sequences: A122416 A122417 A122418 * A122420 A122421 A122422

Keyword

easy,nonn

Author

Vladeta Jovovic, Sep 03 2006

Extensions

More terms from R. J. Mathar, May 18 2007

Status

approved