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%I #22 May 13 2022 18:24:08
%S 1,0,0,0,0,1,6,21,56,126,504,6006,67320,577863,4038034,24975951,
%T 165481680,1553590220,19495772856,249507077436,2910465717648,
%U 31103684847837,326286335505438,3766644374319673,51399738264984648,785038533451101930
%N Expansion of e.g.f. 1 / (1 - x^5 * exp(x) / 5!).
%H Seiichi Manyama, <a href="/A346893/b346893.txt">Table of n, a(n) for n = 0..498</a>
%F a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * binomial(k,5) * a(n-k).
%F a(n) ~ n! / ((1 + LambertW(2^(3/5)*3^(1/5)/5^(4/5))) * 5^(n+1) * LambertW(2^(3/5)*3^(1/5)/5^(4/5))^n). - _Vaclav Kotesovec_, Aug 08 2021
%F a(n) = n! * Sum_{k=0..floor(n/5)} k^(n-5*k)/(120^k * (n-5*k)!). - _Seiichi Manyama_, May 13 2022
%t nmax = 25; CoefficientList[Series[1/(1 - x^5 Exp[x]/5!), {x, 0, nmax}], x] Range[0, nmax]!
%t a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] Binomial[k, 5] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 25}]
%o (PARI) my(x='x+O('x^25)); Vec(serlaplace(1/(1-x^5*exp(x)/5!))) \\ _Michel Marcus_, Aug 06 2021
%o (PARI) a(n) = n!*sum(k=0, n\5, k^(n-5*k)/(120^k*(n-5*k)!)); \\ _Seiichi Manyama_, May 13 2022
%Y Column k=5 of A351703.
%Y Cf. A006153, A346888, A346889, A346890.
%Y Cf. A000389, A145455.
%K nonn
%O 0,7
%A _Ilya Gutkovskiy_, Aug 06 2021
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