A361595 Expansion of e.g.f. exp( (x / (1-x))^3 ) / (1-x).

1, 1, 2, 12, 120, 1320, 15480, 199080, 2862720, 46146240, 826156800, 16212873600, 344741443200, 7875365097600, 192137321376000, 4984375210214400, 136994756496998400, 3976455027389644800, 121533921410994892800, 3900447928934548992000

Offset: 0

Author

Seiichi Manyama, Mar 16 2023

Keywords

• nonn

Crossrefs

Cf. A002720, A361594.

Cf. A361572.

Programs

Mathematica

Table[n! * Sum[Binomial[n,3*k]/k!, {k,0,n/3}], {n,0,20}] (* Vaclav Kotesovec, Mar 17 2023 *)

PARI

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp((x/(1-x))^3)/(1-x)))

Formula

a(n) = n! * Sum_{k=0..floor(n/3)} binomial(n,3*k)/k!.

From Vaclav Kotesovec, Mar 17 2023: (Start)

a(n) = (4*n - 3)*a(n-1) - 3*(n-1)*(2*n - 3)*a(n-2) + 2*(n-2)*(n-1)*(2*n - 3)*a(n-3) - (n-3)^2*(n-2)*(n-1)*a(n-4).

a(n) ~ 3^(-1/8) * exp(-1/4 + 5*3^(-1/4)*n^(1/4)/8 - sqrt(3*n)/2 + 4*3^(-3/4) * n^(3/4) - n) * n^(n + 1/8) / 2 * (1 + (1511/2560)*3^(1/4)/n^(1/4)). (End)