A122418 - OEIS

%I #18 Aug 09 2018 09:45:57

%S 1,0,2,54,2534,186030,19794662,2885980734,552803552534,

%T 134687987183790,40686498089484422,14925683377452413214,

%U 6536580413039406774134,3368723388994026165415950,2018248855531992511720945382,1390953089533285777007059354494,1092714503596231472933813958469334

%N a(n) = Sum_{k=0..n} (k-1)^n*k!*Stirling2(n,k).

%H G. C. Greubel, <a href="/A122418/b122418.txt">Table of n, a(n) for n = 0..229</a>

%F E.g.f.: Sum((exp((n-1)*x)-1)^n, n=0..infinity).

%F 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

%p 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

%t 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 *)

%o (PARI) for(n=0,50, print1(sum(k=0,n, (k-1)^n*k!*stirling(n,k,2)), ", ")) \\ _G. C. Greubel_, Nov 15 2017

%Y Cf. A122419, A122420, A122399.

%K easy,nonn

%O 0,3

%A _Vladeta Jovovic_, Sep 03 2006

%E More terms from _R. J. Mathar_, Feb 10 2007