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A293049 Expansion of e.g.f. exp(x^3/(1 - x)).

%I A293049 #42 Jun 17 2024 10:48:45
%S A293049 1,0,0,6,24,120,1080,10080,100800,1149120,14515200,199584000,
%T A293049 2973801600,47740492800,820928908800,15049152518400,292919058432000,
%U A293049 6031865968128000,130990787582054400,2991455760887193600,71659101232502784000,1796424431562528768000
%N A293049 Expansion of e.g.f. exp(x^3/(1 - x)).
%C A293049 For n > 4, a(n) is a multiple of 10. - _Muniru A Asiru_, Oct 09 2017
%H A293049 Seiichi Manyama, <a href="/A293049/b293049.txt">Table of n, a(n) for n = 0..444</a>
%F A293049 E.g.f.: Product_{i>2} exp(x^i).
%F A293049 a(n) ~ n^(n-1/4) * exp(-5/2 + 2*sqrt(n) - n) / sqrt(2). - _Vaclav Kotesovec_, Sep 30 2017
%F A293049 a(n) = 2*(n-1) * a(n-1) - (n-1)*(n-2) * a(n-2) + 6*binomial(n-1,2) * a(n-3) - 12*binomial(n-1,3) * a(n-4) for n > 3. - _Seiichi Manyama_, Mar 15 2023
%F A293049 From _Seiichi Manyama_, Jun 17 2024: (Start)
%F A293049 a(n) = n! * Sum_{k=0..floor(n/3)} binomial(n-2*k-1,n-3*k)/k!.
%F A293049 a(0) = 1; a(n) = (n-1)! * Sum_{k=3..n} k * a(n-k)/(n-k)!. (End)
%p A293049 a:= proc(n) option remember; `if`(n=0, 1, add(
%p A293049       a(n-j)*binomial(n-1, j-1)*j!, j=3..n))
%p A293049     end:
%p A293049 seq(a(n), n=0..25);  # _Alois P. Heinz_, Sep 30 2017
%p A293049 seq(factorial(k)*coeftayl(exp(x^3/(1-x)), x = 0, k),k=0..50); # _Muniru A Asiru_, Oct 09 2017
%t A293049 CoefficientList[Series[E^(x^3/(1-x)), {x, 0, 20}], x] * Range[0, 20]! (* _Vaclav Kotesovec_, Sep 30 2017 *)
%o A293049 (PARI) x='x+O('x^66); Vec(serlaplace(exp(x^3/(1-x))))
%Y A293049 Column k=2 of A293053.
%Y A293049 E.g.f.: Product_{i>k} exp(x^i): A000262 (k=0), A052845 (k=1), this sequence (k=2), A293050 (k=3).
%Y A293049 Cf. A361533, A361572.
%K A293049 nonn
%O A293049 0,4
%A A293049 _Seiichi Manyama_, Sep 29 2017