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 A305986 Expansion of e.g.f. Product_{k>=1} 1/(1 - (exp(x) - 1)^k/k). 4
 1, 1, 4, 21, 144, 1205, 11908, 135597, 1745488, 25045821, 396249564, 6850289765, 128438323720, 2595394603269, 56224162108468, 1299717221807229, 31931915643021504, 830816659779428525, 22820190255069409804, 659845945466402034165, 20034230527927369097848, 637252918691725377815349 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Stirling transform of A007841. LINKS Vaclav Kotesovec, Table of n, a(n) for n = 0..420 N. J. A. Sloane, Transforms Eric Weisstein's World of Mathematics, Stirling Transform FORMULA E.g.f.: exp(Sum_{k>=1} Sum_{j>=1} (exp(x) - 1)^(j*k)/(k*j^k)). a(n) = Sum_{k=0..n} Stirling2(n,k)*A007841(k). 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 MAPLE b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,       add(combinat[multinomial](n, n-i*j, i\$j)*       b(n-i*j, i-1)*(i-1)!^j, j=0..n/i)))     end: a:= n-> add(Stirling2(n, j)*b(j\$2), j=0..n): seq(a(n), n=0..25);  # Alois P. Heinz, Jun 15 2018 MATHEMATICA nmax = 21; CoefficientList[Series[Product[1/(1 - (Exp[x] - 1)^k/k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]! 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]! 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}] CROSSREFS Cf. A007841, A140585, A167137, A305987. Sequence in context: A006879 A228063 A228111 * A233481 A308337 A307525 Adjacent sequences:  A305983 A305984 A305985 * A305987 A305988 A305989 KEYWORD nonn AUTHOR Ilya Gutkovskiy, Jun 15 2018 STATUS approved

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Last modified January 20 14:02 EST 2020. Contains 331094 sequences. (Running on oeis4.)