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A354120 Expansion of e.g.f. 1/(1 - log(1 + x))^3.

%I A354120 #24 May 16 2025 15:52:26
%S A354120 1,3,9,30,114,492,2388,12912,77016,503112,3570552,27399600,225729360,
%T A354120 1991996640,18690559200,186620451840,1963991600640,21914748541440,
%U A354120 255336518292480,3155705206364160,40209018105116160,547746803311864320,7525926332189130240
%N A354120 Expansion of e.g.f. 1/(1 - log(1 + x))^3.
%C A354120 a(34) is negative. - _Vaclav Kotesovec_, Jun 04 2022
%H A354120 Vaclav Kotesovec, <a href="/A354120/b354120.txt">Table of n, a(n) for n = 0..449</a>
%F A354120 a(n) = (1/2) * Sum_{k=0..n} (k + 2)! * Stirling1(n,k).
%F A354120 a(0) = 1; a(n) = Sum_{k=1..n} (-1)^(k-1) * (2 * k/n + 1) * (k-1)! * binomial(n,k) * a(n-k). - _Seiichi Manyama_, Nov 19 2023
%t A354120 Table[Sum[(k+2)! * StirlingS1[n,k], {k,0,n}]/2, {n,0,35}] (* _Vaclav Kotesovec_, Jun 04 2022 *)
%t A354120 With[{nn=30},CoefficientList[Series[1/(1-Log[1+x])^3,{x,0,nn}],x] Range[0,nn]!] (* _Harvey P. Dale_, May 16 2025 *)
%o A354120 (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-log(1+x))^3))
%o A354120 (PARI) a(n) = sum(k=0, n, (k+2)!*stirling(n, k, 1))/2;
%Y A354120 Cf. A006252, A317280, A354121.
%Y A354120 Cf. A226515, A354122.
%K A354120 sign
%O A354120 0,2
%A A354120 _Seiichi Manyama_, May 17 2022