A360684 - OEIS

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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

(list; graph; refs; listen; history; text; internal format)

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))