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 A305550 Expansion of e.g.f. Product_{k>=1} (1 + (exp(x) - 1)^k). 17
 1, 1, 3, 19, 135, 1171, 12543, 156619, 2185095, 33787171, 579341583, 10927420219, 223956672855, 4940901389971, 116678668726623, 2938719256363819, 78709685812037415, 2234633592020685571, 67005923560416063663, 2114549937496479803419, 70024572874029038582775, 2427790107567416812409971 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Stirling transform of A088311. LINKS Vaclav Kotesovec, Table of n, a(n) for n = 0..400 N. J. A. Sloane, Transforms Eric Weisstein's World of Mathematics, Stirling Transform FORMULA E.g.f.: exp(Sum_{k>=1} (-1)^k*(exp(x) - 1)^k/(k*((exp(x) - 1)^k - 1))). a(n) = Sum_{k=0..n} Stirling2(n,k)*A088311(k). From Vaclav Kotesovec, Jun 17 2018: (Start) a(n) ~ n! * exp(Pi*sqrt(n/(6*log(2))) + (1/log(2) - 1) * Pi^2/48) / (2^(9/4) * 3^(1/4) * n^(3/4) * (log(2))^(n + 1/4)). a(n) ~ sqrt(Pi) * exp(Pi*sqrt(n/(6*log(2))) + (1/log(2) - 1) * Pi^2/48 - n) * n^(n + 1/2) / (2^(7/4) * 3^(1/4) * n^(3/4) * (log(2))^(n + 1/4)). (End) MAPLE b:= proc(n) option remember; `if`(n=0, 1, add(b(n-j)*add(      `if`(d::odd, d, 0), d=numtheory[divisors](j)), j=1..n)/n)     end: a:= n-> add(Stirling2(n, k)*k!*b(k), k=0..n): seq(a(n), n=0..25);  # Alois P. Heinz, Jun 15 2018 MATHEMATICA nmax = 21; CoefficientList[Series[Product[(1 + (Exp[x] - 1)^k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]! nmax = 21; CoefficientList[Series[Exp[Sum[(-1)^k (Exp[x] - 1)^k/(k ((Exp[x] - 1)^k - 1)), {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]! Table[Sum[StirlingS2[n, k] PartitionsQ[k] k!, {k, 0, n}], {n, 0, 21}] CROSSREFS Cf. A000009, A088311, A167137, A306045, A320350. Sequence in context: A074567 A332621 A091346 * A035086 A215852 A105797 Adjacent sequences:  A305547 A305548 A305549 * A305551 A305552 A305553 KEYWORD nonn AUTHOR Ilya Gutkovskiy, Jun 15 2018 STATUS approved

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Last modified July 25 16:39 EDT 2021. Contains 346291 sequences. (Running on oeis4.)