A349882 Expansion of Sum_{k>=0} k^2 * x^k/(1 - k^2 * x).

0, 1, 5, 26, 162, 1267, 12343, 145652, 2036148, 33192789, 622384729, 13263528350, 318121600694, 8517247764135, 252725694989611, 8258153081400856, 295515712276222952, 11523986940937975401, 487562536078882116717, 22291094729329088403298

Offset: 0

Author

Seiichi Manyama, Dec 03 2021

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..339

Crossrefs

Cf. A003101, A234568, A249459, A349863, A359659.

Programs

Mathematica

a[n_] := Sum[If[k == n - k + 1 == 0, 1, k^(2*(n - k + 1))], {k, 0, n}]; Array[a, 20, 0] (* Amiram Eldar, Dec 04 2021 *)

PARI

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

PARI

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

PARI

my(N=20, x='x+O('x^N)); concat(0, Vec(sum(k=1, N, x^k/(1-(k+1)^2*x)))) \\ Seiichi Manyama, Jan 12 2023

Formula

a(n) = Sum_{k=0..n} k^(2*(n-k+1)).

a(n) = A234568(n+1) - 1. - Hugo Pfoertner, Dec 04 2021

a(n) ~ sqrt(Pi) * ((n+1)/LambertW(exp(1)*(n+1)))^(5/2 + 2*n - 2*(n+1)/LambertW(exp(1)*(n+1))) / sqrt(1 + LambertW(exp(1)*(n+1))). - Vaclav Kotesovec, Dec 04 2021

G.f.: Sum_{k>=1} x^k/(1 - (k+1)^2 * x). - Seiichi Manyama, Jan 12 2023