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%I #7 Mar 27 2019 03:53:42
%S 1,1,2,3,14,0,359,-1988,28706,-312210,4387572,-62769366,1006242599,
%T -17203315363,318393704043,-6296931104285,133039045075494,
%U -2986262905171914,71018001954178952,-1783064497977512206,47133484019671647932,-1308274154275749372040,38042727898691562357962
%N Expansion of e.g.f. Product_{k>=1} 1/(1 - log(1 + x)^k/k!).
%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/StirlingTransform.html">Stirling Transform</a>
%F E.g.f.: exp(Sum_{k>=1} Sum_{j>=1} log(1 + x)^(j*k)/(k*(j!)^k)).
%F a(n) = Sum_{k=0..n} Stirling1(n,k)*A005651(k).
%p a:=series(mul(1/(1-log(1+x)^k/k!),k=1..100),x=0,23): seq(n!*coeff(a,x,n),n=0..22); # _Paolo P. Lava_, Mar 26 2019
%t nmax = 22; CoefficientList[Series[Product[1/(1 - Log[1 + x]^k/k!), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
%t nmax = 22; CoefficientList[Series[Exp[Sum[Sum[Log[1 + x]^(j k)/(k (j!)^k), {j, 1, nmax}], {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
%t Table[Sum[StirlingS1[n, k] Total[Apply[Multinomial, IntegerPartitions[k], {1}]], {k, 0, n}], {n, 0, 22}]
%Y Cf. A005651, A140585, A306040.
%K sign
%O 0,3
%A _Ilya Gutkovskiy_, Jun 17 2018
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