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%I #14 Dec 03 2021 09:52:57
%S 0,1,1,-12,50,913,-35093,557048,15098348,-1975843727,103722820509,
%T -396969408196,-704487440688562,106081534307130081,
%U -7995154887120369801,-510404434212739199104,340061796037700870259064,-75820293495741408235957599,8142099722101269515604494937
%N Expansion of Sum_{k>=0} k * x^k/(1 + k^3 * x).
%F a(n) = Sum_{k=0..n} (-k^3)^(n-k) * k.
%t a[n_] := Sum[If[k == n - k == 0, 1, (-k^3)^(n-k)] * k, {k, 0, n}]; Array[a, 19, 0] (* _Amiram Eldar_, Dec 03 2021 *)
%o (PARI) a(n, s=1, t=3) = sum(k=0, n, (-k^t)^(n-k)*k^s);
%o (PARI) my(N=20, x='x+O('x^N)); concat(0, Vec(sum(k=0, N, k*x^k/(1+k^3*x))))
%Y Cf. A349859, A349861.
%K sign
%O 0,4
%A _Seiichi Manyama_, Dec 02 2021
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