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A320344 Expansion of e.g.f. log(1 + x)/(1 - log(1 + x))^2.

%I A320344 #36 Apr 22 2022 15:57:04
%S A320344 0,1,3,8,26,94,406,1896,10440,59472,405264,2673648,22396128,160828368,
%T A320344 1704287568,11993279232,177349981824,957018589056,25766036316288,
%U A320344 33555346603776,5403108443855616,-28811285794990080,1643455634670489600,-21001090458387594240,692074413969784289280
%N A320344 Expansion of e.g.f. log(1 + x)/(1 - log(1 + x))^2.
%H A320344 Seiichi Manyama, <a href="/A320344/b320344.txt">Table of n, a(n) for n = 0..451</a>
%F A320344 a(n) = Sum_{k=0..n} Stirling1(n,k)*A001563(k).
%F A320344 E.g.f.: Sum_{k>=0} k * log(1+x)^k. - _Seiichi Manyama_, Apr 22 2022
%p A320344 seq(n!*coeff(series(log(1+x)/(1-log(1+x))^2,x=0,25),x,n),n=0..24); # _Paolo P. Lava_, Jan 29 2019
%t A320344 nmax = 24; CoefficientList[Series[Log[1 + x]/(1 - Log[1 + x])^2, {x, 0, nmax}], x] Range[0, nmax]!
%t A320344 Table[Sum[StirlingS1[n, k] k k!, {k, 0, n}], {n, 0, 24}]
%o A320344 (PARI) my(N=40, x='x+O('x^N)); concat(0, Vec(serlaplace(sum(k=0, N, k*log(1+x)^k)))) \\ _Seiichi Manyama_, Apr 22 2022
%Y A320344 Cf. A001563, A048994, A069321, A215916, A317280, A351280.
%K A320344 sign
%O A320344 0,3
%A A320344 _Ilya Gutkovskiy_, Jan 22 2019