A122420 - OEIS

%I #13 Aug 09 2018 09:46:31

%S 1,0,1,10,120,1778,31685,661940,15882128,430607370,13022755068,

%T 434697574538,15875944361864,629756003982336,26963278837704185,

%U 1239382820431888898,60875147436141987437,3181961834442383306068

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

%F 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.

%F 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

%p 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

%t 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 *)

%Y Cf. A104209.

%K easy,nonn

%O 0,4

%A _Vladeta Jovovic_, Sep 03 2006

%E More terms from _R. J. Mathar_, May 18 2007