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A192554 a(n) = Sum_{k=0..n} abs(Stirling1(n,k))*(-1)^(n-k)*k!^2.

%I A192554 #21 Apr 23 2022 01:35:13
%S A192554 1,1,3,26,398,9724,344236,16663968,1056631824,84962783664,
%T A192554 8446120969104,1016998946575776,145848462866589600,
%U A192554 24562489788256472064,4799789988678066147840,1077128972416478325901824,275111625956753684599202304
%N A192554 a(n) = Sum_{k=0..n} abs(Stirling1(n,k))*(-1)^(n-k)*k!^2.
%H A192554 Seiichi Manyama, <a href="/A192554/b192554.txt">Table of n, a(n) for n = 0..253</a>
%F A192554 a(n) = Sum_{k=0..n} Stirling1(n,k) * k!^2. - _Vaclav Kotesovec_, Jul 05 2021
%F A192554 a(n) ~ exp(-1/2) * n!^2. - _Vaclav Kotesovec_, Jul 05 2021
%F A192554 E.g.f.: Sum_{k>=0} k! * log(1+x)^k. - _Seiichi Manyama_, Apr 22 2022
%t A192554 Table[Sum[Abs[StirlingS1[n,k]](-1)^(n-k)k!^2,{k,0,n}],{n,0,100}]
%t A192554 Table[Sum[StirlingS1[n,k] * k!^2, {k, 0, n}], {n, 0, 20}] (* _Vaclav Kotesovec_, Jul 05 2021 *)
%o A192554 (Maxima) makelist(sum(abs(stirling1(n,k))*(-1)^(n-k)*k!^2,k,0,n),n,0,24);
%o A192554 (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 A192554 Cf. A006252, A320502, A351280.
%Y A192554 Cf. A064618.
%K A192554 nonn
%O A192554 0,3
%A A192554 _Emanuele Munarini_, Jul 04 2011