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A302547 Expansion of e.g.f. -log(1 - log(1 + x))/(1 - log(1 + x)).

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