A360727 Expansion of Sum_{k>=0} (k * x * (1 + x^2))^k.

1, 1, 4, 28, 264, 3206, 47684, 839249, 17058688, 393216567, 10134918592, 288815780665, 9016571143680, 306027510946208, 11219450971161024, 441846991480590475, 18602901833071633792, 833832341625621777368, 39642569136740054367808

Offset: 0

Author

Seiichi Manyama, Feb 18 2023

Keywords

• nonn

Links

• Winston de Greef, Table of n, a(n) for n = 0..385

Crossrefs

Cf. A360611, A360728.

Cf. A360730.

Programs

Mathematica

nmax = 20; CoefficientList[1 + Series[Sum[(k*x*(1 + x^2))^k, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Feb 18 2023 *)

PARI

my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, (k*x*(1+x^2))^k))

PARI

a(n) = sum(k=0, n\3, (n-2*k)^(n-2*k)*binomial(n-2*k, k));

Formula

a(n) = Sum_{k=0..floor(n/3)} (n-2*k)^(n-2*k) * binomial(n-2*k,k).

a(n) ~ n^n * (1 + exp(-2)/n + exp(-4)/(2*n^2)). - Vaclav Kotesovec, Feb 18 2023