A361594 Expansion of e.g.f. exp( (x / (1-x))^2 ) / (1-x).

1, 1, 4, 24, 180, 1620, 17040, 204960, 2770320, 41504400, 681791040, 12173293440, 234555773760, 4847900016960, 106932303878400, 2506094618227200, 62165827044921600, 1626693694039814400, 44767280999939097600, 1292282276155782912000

Offset: 0

Author

Seiichi Manyama, Mar 16 2023

Keywords

• nonn

Crossrefs

Cf. A002720, A361595.

Cf. A052887.

Programs

Mathematica

Table[n! * Sum[Binomial[n,2*k]/k!, {k,0,n/2}], {n,0,20}] (* Vaclav Kotesovec, Mar 17 2023 *)
With[{nn=20},CoefficientList[Series[Exp[(x/(1-x))^2]/(1-x),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Oct 29 2023 *)

PARI

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp((x/(1-x))^2)/(1-x)))

Formula

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

From Vaclav Kotesovec, Mar 17 2023: (Start)

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

a(n) ~ 2^(-1/6) * 3^(-1/2) * exp(1/3 - 2^(-1/3)*n^(1/3) + 3*2^(-2/3)*n^(2/3) - n) * n^(n + 1/6) * (1 + 11*2^(1/3)/(27*n^(1/3)) - 79/(3645*2^(1/3)*n^(2/3))). (End)