A349902 -id:A349902 - cp's OEIS frontend

A349857 Expansion of Sum_{k>=0} x^k/(1 + k^3 * x).

1, 1, 0, -6, 37, 155, -11616, 251940, 783641, -454238419, 29895012768, -757531311386, -106105977022243, 21452688824818775, -2105573104903303616, 16702280440994303008, 48278492787774402969521, -13301912828187822051695559, 1964564462643243537548661568

Offset: 0

Seiichi Manyama, Dec 02 2021

sign

Seiichi Manyama, Table of n, a(n) for n = 0..247

Cf. A349856, A349858, A349902.

a[n_] := Sum[If[k == n - k == 0, 1, (-k^3)^(n-k)], {k, 0, n}]; Array[a, 19, 0] (* Amiram Eldar, Dec 03 2021 *)

a(n, s=0, t=3) = sum(k=0, n, (-k^t)^(n-k)*k^s);

my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, x^k/(1+k^3*x)))

a(n) = Sum_{k=0..n} (-k^3)^(n-k).

A349889 a(n) = Sum_{k=0..n} (-1)^(n-k) * k^(2*n).

Original entry on oeis.org

1, 1, 15, 666, 59230, 8775075, 1948891581, 605698755508, 250914820143996, 133610836793706405, 88919025666286620475, 72317513878698256697166, 70571883548815735717843290, 81383769918571603591381635271

Offset: 0

Seiichi Manyama, Dec 04 2021

nonn

Cf. A120485, A249459, A349884, A349891, A349902.

Join[{1},Table[Sum[(-1)^(n-k) k^(2n),{k,0,n}],{n,20}]] (* Harvey P. Dale, Nov 19 2023 *)

a(n) = sum(k=0, n, (-1)^(n-k)*k^(2*n));

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

G.f.: Sum_{k>=0} (k^2 * x)^k/(1 + k^2 * x).

a(n) ~ 1/(1 + exp(-2)) * n^(2*n). - Vaclav Kotesovec, Dec 10 2021

cp's OEIS Frontend

A349857 Expansion of Sum_{k>=0} x^k/(1 + k^3 * x).

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A349889 a(n) = Sum_{k=0..n} (-1)^(n-k) * k^(2*n).

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