A192554 - OEIS

%I #21 Apr 23 2022 01:35:13

%S 1,1,3,26,398,9724,344236,16663968,1056631824,84962783664,

%T 8446120969104,1016998946575776,145848462866589600,

%U 24562489788256472064,4799789988678066147840,1077128972416478325901824,275111625956753684599202304

%N a(n) = Sum_{k=0..n} abs(Stirling1(n,k))*(-1)^(n-k)*k!^2.

%H Seiichi Manyama, <a href="/A192554/b192554.txt">Table of n, a(n) for n = 0..253</a>

%F a(n) = Sum_{k=0..n} Stirling1(n,k) * k!^2. - _Vaclav Kotesovec_, Jul 05 2021

%F a(n) ~ exp(-1/2) * n!^2. - _Vaclav Kotesovec_, Jul 05 2021

%F E.g.f.: Sum_{k>=0} k! * log(1+x)^k. - _Seiichi Manyama_, Apr 22 2022

%t Table[Sum[Abs[StirlingS1[n,k]](-1)^(n-k)k!^2,{k,0,n}],{n,0,100}]

%t Table[Sum[StirlingS1[n,k] * k!^2, {k, 0, n}], {n, 0, 20}] (* _Vaclav Kotesovec_, Jul 05 2021 *)

%o (Maxima) makelist(sum(abs(stirling1(n,k))*(-1)^(n-k)*k!^2,k,0,n),n,0,24);

%o (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, k!*log(1+x)^k))) \\ _Seiichi Manyama_, Apr 22 2022

%Y Cf. A006252, A320502, A351280.

%Y Cf. A064618.

%K nonn

%O 0,3

%A _Emanuele Munarini_, Jul 04 2011