A389992 Expansion of e.g.f. exp( x/(1-x)^2 ) / (1-x)^3.

1, 4, 23, 172, 1573, 16916, 208339, 2883868, 44237897, 743819428, 13588696111, 267788968844, 5658385489453, 127541852311732, 3053231291481323, 77330250770320636, 2065194851018797969, 57983115305964783428, 1706939688831615952327, 52562564587644514931308

Offset

0,2

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..430

Formula

a(n) = n! * Sum_{k=0..n} binomial(n+k+2,n-k)/k!.

From Vaclav Kotesovec, Oct 21 2025: (Start)

a(n) = (3*n+1)*a(n-1) - (n-1)*(3*n-1)*a(n-2) + (n-2)*(n-1)*n*a(n-3).

a(n) ~ 2^(-5/6) * 3^(-1/2) * exp(3*n^(2/3)/2^(2/3) - n - 1/12) * n^(n + 5/6) * (1 + 3/(2^(2/3)*n^(1/3)) + 623/(360*2^(1/3)*n^(2/3)) + 223/(240*n)). (End)

Mathematica

nmax = 20; CoefficientList[Series[E^(x/(1-x)^2)/(1-x)^3, {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Oct 21 2025 *)

Prog

(PARI) a(n) = n!*sum(k=0, n, binomial(n+k+2, n-k)/k!);

Crossrefs

Cf. A343884, A361598.

Cf. A052852, A351767.

Sequence in context: A317276 A113869 A084357 * A360868 A075729 A328006

Adjacent sequences: A389989 A389990 A389991 * A389993 A389994 A389995

Keyword

nonn,easy

Author

Seiichi Manyama, Oct 21 2025

Status

approved