A122420 Number of labeled directed multigraphs with n arcs and with no vertex of indegree 0.

1, 0, 1, 10, 120, 1778, 31685, 661940, 15882128, 430607370, 13022755068, 434697574538, 15875944361864, 629756003982336, 26963278837704185, 1239382820431888898, 60875147436141987437, 3181961834442383306068

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.: A(x/(1-x)) where A(x) is g.f. for A122419.

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 = 0.1221803955695846906452721220983425... . - Vaclav Kotesovec, May 07 2014

Maple

A122418 := proc(n) option remember ; add( combinat[stirling2](n, k)*(k-1)^n*k!, k=0..n) ; end: A122420 := proc(n) option remember ; add( abs(combinat[stirling1](n, k))*A122418(k), k=0..n)/n! ; end: for n from 0 to 30 do printf("%d, ", A122420(n)) ; od ; # R. J. Mathar, May 18 2007

Mathematica

Table[1/n!*Sum[Abs[StirlingS1[n, k]]*Sum[(m-1)^k*m!*StirlingS2[k, m], {m, 0, k}], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, May 07 2014 *)

Crossrefs

Cf. A104209.

Sequence in context: A034255 A363310 A051582 * A069671 A138445 A138496

Adjacent sequences: A122417 A122418 A122419 * A122421 A122422 A122423

Keyword

easy,nonn

Author

Vladeta Jovovic, Sep 03 2006

Extensions

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

Status

approved