A360647 Expansion of Sum_{k>=0} (k^2 * x * (1 + x))^k.

1, 1, 17, 761, 67739, 10029956, 2226004406, 691381685259, 286255287677425, 152360721379689043, 101358756787489940837, 82408168580060017122144, 80396790074312939684672316, 92691781529853274368541343021

Offset: 0

Author

Seiichi Manyama, Feb 15 2023

Keywords

• nonn

Crossrefs

Cf. A323280, A360611.

Programs

Mathematica

Flatten[{1, Table[Sum[Binomial[n-k,k] * (n-k)^(2*(n-k)), {k,0,n/2}], {n,1,30}]}] (* Vaclav Kotesovec, Feb 16 2023 *)

PARI

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

PARI

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

Formula

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

a(n) ~ n^(2*n). - Vaclav Kotesovec, Feb 16 2023