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Expansion of Sum_{k>=0} x^k/(1 + k^3 * x).
5

%I #20 Dec 11 2021 04:29:30

%S 1,1,0,-6,37,155,-11616,251940,783641,-454238419,29895012768,

%T -757531311386,-106105977022243,21452688824818775,

%U -2105573104903303616,16702280440994303008,48278492787774402969521,-13301912828187822051695559,1964564462643243537548661568

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

%H Seiichi Manyama, <a href="/A349857/b349857.txt">Table of n, a(n) for n = 0..247</a>

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

%t 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 *)

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

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

%Y Cf. A349856, A349858, A349902.

%K sign

%O 0,4

%A _Seiichi Manyama_, Dec 02 2021