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