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%I #33 Mar 25 2023 06:46:25
%S 1,4,25,214,2293,29176,427189,7049890,129178249,2597880268,
%T 56815155121,1341068392654,33951269718205,917020113259264,
%U 26305693331946253,798293630021120986,25540244079135784849,858854698277997113620,30274382852181639467209
%N Expansion of e.g.f. exp( x/(1-x)^3 ) / (1-x)^3.
%H Winston de Greef, <a href="/A351767/b351767.txt">Table of n, a(n) for n = 0..420</a>
%F a(n) = n! * Sum_{k=0..n} binomial(n+2*k+2,n-k)/k! = Sum_{k=0..n} (n+2*k+2)!/(3*k+2)! * binomial(n,k).
%F From _Vaclav Kotesovec_, Mar 25 2023: (Start)
%F a(n) = 4*n*a(n-1) - (n-1)*(6*n - 5)*a(n-2) + (n-2)*(n-1)*(4*n - 3)*a(n-3) - (n-3)*(n-2)*(n-1)^2*a(n-4).
%F a(n) ~ exp(-1/27 - 3^(-5/4)*n^(1/4)/8 + sqrt(n/3)/2 + 4*3^(-3/4)*n^(3/4) - n) * n^(n + 5/8) / (2 * 3^(5/8)) * (1 + 91837/69120 * 3^(1/4)/n^(1/4)). (End)
%t Table[n!*Sum[Binomial[n + 2*k + 2, n - k]/k!, {k, 0, n}], {n, 0, 20}] (* _Vaclav Kotesovec_, Mar 25 2023 *)
%o (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(x/(1-x)^3)/(1-x)^3))
%o (PARI) a(n) = n! * sum(k=0, n, binomial(n+2*k+2,n-k)/k!); \\ _Winston de Greef_, Mar 18 2023
%Y Column k=3 of A361616.
%Y Cf. A052852, A361599.
%K nonn
%O 0,2
%A _Seiichi Manyama_, Mar 18 2023
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