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Expansion of Sum_{k>=0} k^3 * x^k/(1 + k * x).
4

%I #19 Dec 03 2021 09:54:01

%S 0,1,7,12,14,49,13,8,596,-1967,4011,9764,-128878,664545,-1837695,

%T -2388448,67004968,-478198239,1994890287,-1669470404,-56929813514,

%U 615188040657,-3794477505067,12028579020088,50780206473820,-1172949397923535,10766410530764819,-61183127006113196

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

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

%t a[n_] := -Sum[(-k)^(n - k + 3), {k, 0, n}]; Array[a, 28, 0] (* _Amiram Eldar_, Dec 02 2021 *)

%o (PARI) a(n, s=3, t=1) = sum(k=0, n, (-k^t)^(n-k)*k^s);

%o (PARI) my(N=40, x='x+O('x^N)); concat(0, Vec(sum(k=0, N, k^3*x^k/(1+k*x))))

%Y Cf. A038125, A349852, A349853, A349855.

%K sign

%O 0,3

%A _Seiichi Manyama_, Dec 02 2021