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%I #8 Apr 13 2019 04:26:19
%S 1,1,4,21,146,1240,12479,144970,1908682,28079550,456458832,8125189974,
%T 157190542607,3284222304545,73705849847317,1768479436456975,
%U 45180024672023814,1224529894981726614,35096983241255523572,1060703070504583747430,33714045363258013414692
%N Expansion of e.g.f. Product_{k>=1} 1/(1 - log(1/(1 - x))^k/k!).
%F E.g.f.: exp(Sum_{k>=1} Sum_{j>=1} log(1/(1 - x))^(j*k)/(k*(j!)^k)).
%F a(n) = Sum_{k=0..n} |Stirling1(n,k)|*A005651(k).
%F a(n) ~ c * sqrt(2*Pi) * n^(n + 1/2) / (exp(1) - 1)^(n+1), where c = A247551 = Product_{k>=2} 1/(1-1/k!). - _Vaclav Kotesovec_, Apr 13 2019
%t nmax = 20; CoefficientList[Series[Product[1/(1 - Log[1/(1 - x)]^k/k!), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
%t nmax = 20; CoefficientList[Series[Exp[Sum[Sum[Log[1/(1 - x)]^(j k)/(k (j!)^k), {j, 1, nmax}], {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
%t Table[Sum[Abs[StirlingS1[n, k]] Total[Apply[Multinomial, IntegerPartitions[k], {1}]], {k, 0, n}], {n, 0, 20}]
%Y Cf. A005651, A140585, A306039, A320349.
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, Apr 12 2019
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