%I #11 Jun 29 2018 21:47:17
%S 0,1,2,4,11,33,131,516,2810,12934,97870,447940,5308112,16394116,
%T 450505844,-315178912,60774618672,-394330113648,12662225550288,
%U -157622647720032,3766647294946944,-64679214198647520,1475157821754785184,-30431206030329719424,719032203373502252160
%N Expansion of e.g.f. -log(1 - log(1 + x))/(1 - log(1 + x)).
%H Alois P. Heinz, <a href="/A302547/b302547.txt">Table of n, a(n) for n = 0..451</a>
%F a(n) = Sum_{k=1..n} Stirling1(n,k)*H(k)*k!, where H(k) is the k-th harmonic number.
%e E.g.f.: A(x) = x + 2*x^2/2! + 4*x^3/3! + 11*x^4/4! + 33*x^5/5! + 131*x^6/6! + ...
%p H:= proc(n) H(n):= 1/n +`if`(n=1, 0, H(n-1)) end:
%p a:= n-> add(Stirling1(n, k)*H(k)*k!, k=1..n):
%p seq(a(n), n=0..27); # _Alois P. Heinz_, Jun 21 2018
%t nmax = 24; CoefficientList[Series[-Log[1 - Log[1 + x]]/(1 - Log[1 + x]), {x, 0, nmax}], x] Range[0, nmax]!
%t Table[Sum[StirlingS1[n, k] HarmonicNumber[k] k!, {k, 0, n}], {n, 0, 24}]
%Y Cf. A000254, A001008, A002805, A006252, A073596, A089064, A222058, A300490, A302548.
%K sign
%O 0,3
%A _Ilya Gutkovskiy_, Jun 20 2018