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A293050 Expansion of e.g.f. exp(x^4/(1 - x)).

%I A293050 #29 Jun 17 2024 10:48:41
%S A293050 1,0,0,0,24,120,720,5040,60480,725760,9072000,119750400,1756339200,
%T A293050 28021593600,479480601600,8717829120000,168254102016000,
%U A293050 3438311804928000,74160828758016000,1682757222322176000,40061786401308672000,998402161605488640000
%N A293050 Expansion of e.g.f. exp(x^4/(1 - x)).
%H A293050 Seiichi Manyama, <a href="/A293050/b293050.txt">Table of n, a(n) for n = 0..444</a>
%F A293050 E.g.f.: Product_{i>3} exp(x^i).
%F A293050 From _Vaclav Kotesovec_, Sep 30 2017: (Start)
%F A293050 a(n) = 2*(n-1)*a(n-1) - (n-2)*(n-1)*a(n-2) + 4*(n-3)*(n-2)*(n-1)*a(n-4) - 3*(n-4)*(n-3)*(n-2)*(n-1)*a(n-5).
%F A293050 a(n) ~ n^(n-1/4) * exp(-7/2 + 2*sqrt(n) - n) / sqrt(2).
%F A293050 (End)
%F A293050 From _Seiichi Manyama_, Jun 17 2024: (Start)
%F A293050 a(n) = n! * Sum_{k=0..floor(n/4)} binomial(n-3*k-1,n-4*k)/k!.
%F A293050 a(0) = 1; a(n) = (n-1)! * Sum_{k=4..n} k * a(n-k)/(n-k)!. (End)
%p A293050 a:= proc(n) option remember; `if`(n=0, 1, add(
%p A293050       a(n-j)*binomial(n-1, j-1)*j!, j=4..n))
%p A293050     end:
%p A293050 seq(a(n), n=0..23);  # _Alois P. Heinz_, Sep 29 2017
%t A293050 a[n_] := a[n] = If[n==0, 1, Sum[a[n-j] Binomial[n-1, j-1] j!, {j, 4, n}]];
%t A293050 a /@ Range[0, 23] (* _Jean-François Alcover_, Dec 21 2020, after _Alois P. Heinz_ *)
%o A293050 (PARI) x='x+O('x^66); Vec(serlaplace(exp(x^4/(1-x))))
%Y A293050 Column k=3 of A293053.
%Y A293050 E.g.f.: Product_{i>k} exp(x^i): A000262 (k=0), A052845 (k=1), A293049 (k=2), this sequence (k=3).
%Y A293050 Cf. A361545, A361576.
%K A293050 nonn
%O A293050 0,5
%A A293050 _Seiichi Manyama_, Sep 29 2017