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A308862 Expansion of e.g.f. 1/(1 - x*(1 + 3*x + x^2)*exp(x)).

%I A308862 #14 Mar 10 2022 07:57:09
%S A308862 1,1,10,81,976,14505,258456,5377897,127852096,3419620209,101625743080,
%T A308862 3322169384721,118475520287136,4577175039397753,190436902905933880,
%U A308862 8489222610046324665,403657900923994965376,20393319895130130117729,1090902632352025316904648
%N A308862 Expansion of e.g.f. 1/(1 - x*(1 + 3*x + x^2)*exp(x)).
%H A308862 Seiichi Manyama, <a href="/A308862/b308862.txt">Table of n, a(n) for n = 0..381</a>
%F A308862 E.g.f.: 1 / (1 - Sum_{k>=1} k^3*x^k/k!).
%F A308862 a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * k^3 * a(n-k).
%F A308862 a(n) ~ n! / (r^(n+1) * exp(r) * (1 + 7*r + 6*r^2 + r^3)), where r = 0.33649177041401456061485914122406146158245451810028937972189... is the root of the equation exp(r)*r*(1 + 3*r + r^2) = 1. - _Vaclav Kotesovec_, Jun 29 2019
%t A308862 nmax = 18; CoefficientList[Series[1/(1 - x (1 + 3 x + x^2) Exp[x]), {x, 0, nmax}], x] Range[0, nmax]!
%t A308862 a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] k^3 a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 18}]
%o A308862 (PARI) my(x='x+O('x^25)); Vec(serlaplace(1/(1 - x*(1 + 3*x + x^2)*exp(x)))) \\ _Michel Marcus_, Mar 10 2022
%Y A308862 Cf. A000578, A006153, A144109, A279358, A302870, A308861.
%K A308862 nonn
%O A308862 0,3
%A A308862 _Ilya Gutkovskiy_, Jun 29 2019