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%I #20 Dec 03 2021 09:53:51
%S 0,1,15,50,76,203,335,-84,2696,-3011,-8433,130606,-662348,1840439,
%T 2391823,-67000872,478203152,-1994884455,1669477263,56929821514,
%U -615188031396,3794477515715,-12028579007921,-50780206459996,1172949397939160,-10766410530747243
%N Expansion of Sum_{k>=0} k^4 * x^k/(1 + k * x).
%F a(n) = Sum_{k=0..n} (-k)^(n-k+4).
%t a[n_] := Sum[(-k)^(n - k + 4), {k, 0, n}]; Array[a, 26, 0] (* _Amiram Eldar_, Dec 02 2021 *)
%o (PARI) a(n, s=4, 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^4*x^k/(1+k*x))))
%Y Cf. A038125, A349852, A349853, A349854.
%K sign
%O 0,3
%A _Seiichi Manyama_, Dec 02 2021
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