A349863 Expansion of Sum_{k>=0} k^2 * x^k/(1 + k^2 * x).

0, 1, 3, -6, -2, 243, -2031, 3796, 187212, -3860139, 36467311, 284357502, -21796446486, 538332144295, -5605176351651, -182065102478856, 12963817679287960, -422751776737348503, 5483284328996107803, 327213964461103956802, -30082452646697648945898

Offset: 0

Author

Seiichi Manyama, Dec 02 2021

Keywords

• sign

Crossrefs

Cf. A349852.

Programs

Mathematica

a[n_] := Sum[If[k == n - k == 0, 1, (-k^2)^(n - k)] * k^2, {k, 0, n}]; Array[a, 21, 0] (* Amiram Eldar, Dec 03 2021 *)

PARI

a(n, s=2, t=2) = sum(k=0, n, (-k^t)^(n-k)*k^s);

PARI

my(N=40, x='x+O('x^N)); concat(0, Vec(sum(k=0, N, k^2*x^k/(1+k^2*x))))

Formula

a(n) = Sum_{k=0..n} (-k^2)^(n-k) * k^2.