A349884 Expansion of Sum_{k>=0} (k * x)^k/(1 + k^2 * x).

1, 1, 3, 12, 76, 961, 15407, 221528, 3260936, 80774113, 2462081967, 50963779604, 922244742292, 61063845514113, 2868669700179871, 2019727494212912, -47889136910252848, 461395118866593115713, 5781219348638565771423, -2108738296748190078596084

Offset: 0

Author

Seiichi Manyama, Dec 03 2021

Keywords

• sign

Crossrefs

Cf. A120485, A349885.

Cf. A349856, A349859, A349863.

Programs

Mathematica

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

PARI

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

PARI

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

Formula

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