A360684 Expansion of Sum_{k>=0} (x * (1 + k^2 * x))^k.

1, 1, 2, 9, 44, 308, 2391, 22851, 241570, 2937179, 39192998, 579482352, 9328260061, 162563246381, 3062996934322, 61499850730949, 1327236820161040, 30176760155713420, 733829463528115523, 18639130961053854975, 504241689606231891890

Offset

0,3

Links

Table of n, a(n) for n=0..20.

Formula

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

a(n) ~ (exp(exp(1)) + (-1)^n * exp(-exp(1))) * n^n / 2^(n+1). - Vaclav Kotesovec, Feb 16 2023

Mathematica

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

Prog

(PARI) my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, (x*(1+k^2*x))^k))
(PARI) a(n) = sum(k=0, n\2, (n-k)^(2*k)*binomial(n-k, k));

Crossrefs

Cf. A355471, A360592, A360647.

Sequence in context: A184932 A073478 A336400 * A357682 A322613 A318913

Adjacent sequences: A360681 A360682 A360683 * A360685 A360686 A360687

Keyword

nonn

Author

Seiichi Manyama, Feb 16 2023

Status

approved