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%I #28 Dec 03 2021 09:54:33
%S 0,1,1,0,2,1,-5,20,-28,-47,525,-2056,3902,9633,-129033,664364,
%T -1837904,-2388687,67004697,-478198544,1994889946,-1669470783,
%U -56929813933,615188040196,-3794477505572,12028579019537,50780206473221,-1172949397924184,10766410530764118
%N Expansion of Sum_{k>=0} k * x^k/(1 + k * x).
%F a(n) = -Sum_{k=0..n} (-k)^(n-k+1).
%t a[n_] := -Sum[(-k)^(n - k + 1), {k, 0, n}]; Array[a, 29, 0] (* _Amiram Eldar_, Dec 02 2021 *)
%o (PARI) a(n, s=1, 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*x^k/(1+k*x))))
%Y Cf. A038125, A349853, A349854, A349855, A349859, A349860, A349861.
%K sign
%O 0,5
%A _Seiichi Manyama_, Dec 02 2021
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