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A305986 Expansion of e.g.f. Product_{k>=1} 1/(1 - (exp(x) - 1)^k/k).

%I A305986 #14 Feb 16 2025 08:33:54
%S A305986 1,1,4,21,144,1205,11908,135597,1745488,25045821,396249564,6850289765,
%T A305986 128438323720,2595394603269,56224162108468,1299717221807229,
%U A305986 31931915643021504,830816659779428525,22820190255069409804,659845945466402034165,20034230527927369097848,637252918691725377815349
%N A305986 Expansion of e.g.f. Product_{k>=1} 1/(1 - (exp(x) - 1)^k/k).
%C A305986 Stirling transform of A007841.
%H A305986 Vaclav Kotesovec, <a href="/A305986/b305986.txt">Table of n, a(n) for n = 0..420</a>
%H A305986 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%H A305986 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/StirlingTransform.html">Stirling Transform</a>
%F A305986 E.g.f.: exp(Sum_{k>=1} Sum_{j>=1} (exp(x) - 1)^(j*k)/(k*j^k)).
%F A305986 a(n) = Sum_{k=0..n} Stirling2(n,k)*A007841(k).
%F A305986 a(n) ~ c * n! * n / log(2)^n, where c = exp(-gamma) / (4*log(2)^2) = 0.29215... and gamma is the Euler-Mascheroni constant A001620. - _Vaclav Kotesovec_, Jul 23 2019
%p A305986 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A305986       add(combinat[multinomial](n, n-i*j, i$j)*
%p A305986       b(n-i*j, i-1)*(i-1)!^j, j=0..n/i)))
%p A305986     end:
%p A305986 a:= n-> add(Stirling2(n, j)*b(j$2), j=0..n):
%p A305986 seq(a(n), n=0..25);  # _Alois P. Heinz_, Jun 15 2018
%t A305986 nmax = 21; CoefficientList[Series[Product[1/(1 - (Exp[x] - 1)^k/k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
%t A305986 nmax = 21; CoefficientList[Series[Exp[Sum[Sum[(Exp[x] - 1)^(j k)/(k j^k), {j, 1, nmax}], {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
%t A305986 b[0] = 1; b[n_] := b[n] = Sum[(n - 1)!/(n - k)! DivisorSum[k, #^(1 - k/#) &] b[n - k], {k, 1, n}]; a[n_] := a[n] = Sum[StirlingS2[n, k] b[k], {k, 0, n}]; Table[a[n], {n, 0, 21}]
%Y A305986 Cf. A007841, A140585, A167137, A305987.
%K A305986 nonn
%O A305986 0,3
%A A305986 _Ilya Gutkovskiy_, Jun 15 2018