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A308861 Expansion of e.g.f. 1/(1 - x*(1 + x)*exp(x)).

%I A308861 #17 Mar 10 2022 07:57:14
%S A308861 1,1,6,39,352,3965,53556,844123,15204960,308118105,6937562980,
%T A308861 171826160231,4642588564032,135891789038629,4283619809941668,
%U A308861 144674451274329075,5211965027738046016,199498704931954788785,8085413817213212761668,345895984008645703002559
%N A308861 Expansion of e.g.f. 1/(1 - x*(1 + x)*exp(x)).
%H A308861 Seiichi Manyama, <a href="/A308861/b308861.txt">Table of n, a(n) for n = 0..396</a>
%F A308861 E.g.f.: 1 / (1 - Sum_{k>=1} k^2*x^k/k!).
%F A308861 a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * k^2 * a(n-k).
%F A308861 a(n) ~ n! / (r^(n+1) * exp(r) * (1 + 3*r + r^2)), where r = A201941 = 0.44413022882396659058546632949098466707932096994213775695918... is the root of the equation exp(r)*r*(1 + r) = 1. - _Vaclav Kotesovec_, Jun 29 2019
%t A308861 nmax = 19; CoefficientList[Series[1/(1 - x (1 + x) Exp[x]), {x, 0, nmax}], x] Range[0, nmax]!
%t A308861 a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] k^2 a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 19}]
%o A308861 (PARI) my(x='x+O('x^25)); Vec(serlaplace(1/(1 - x*(1 + x)*exp(x)))) \\ _Michel Marcus_, Mar 10 2022
%Y A308861 Cf. A000290, A006153, A033453, A033462, A302189, A308862.
%K A308861 nonn
%O A308861 0,3
%A A308861 _Ilya Gutkovskiy_, Jun 29 2019