A351767 Expansion of e.g.f. exp( x/(1-x)^3 ) / (1-x)^3.

1, 4, 25, 214, 2293, 29176, 427189, 7049890, 129178249, 2597880268, 56815155121, 1341068392654, 33951269718205, 917020113259264, 26305693331946253, 798293630021120986, 25540244079135784849, 858854698277997113620, 30274382852181639467209

Offset

0,2

Links

Winston de Greef, Table of n, a(n) for n = 0..420

Formula

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).

From Vaclav Kotesovec, Mar 25 2023: (Start)

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).

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)

Mathematica

Table[n!*Sum[Binomial[n + 2*k + 2, n - k]/k!, {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Mar 25 2023 *)

Prog

(PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(x/(1-x)^3)/(1-x)^3))
(PARI) a(n) = n! * sum(k=0, n, binomial(n+2*k+2, n-k)/k!); \\ Winston de Greef, Mar 18 2023

Crossrefs

Column k=3 of A361616.

Cf. A052852, A361599.

Sequence in context: A305323 A215094 A047733 * A198198 A007830 A305404

Adjacent sequences: A351764 A351765 A351766 * A351768 A351769 A351770

Keyword

nonn

Author

Seiichi Manyama, Mar 18 2023

Status

approved