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

1, 1, 4, 21, 144, 1205, 11908, 135597, 1745488, 25045821, 396249564, 6850289765, 128438323720, 2595394603269, 56224162108468, 1299717221807229, 31931915643021504, 830816659779428525, 22820190255069409804, 659845945466402034165, 20034230527927369097848, 637252918691725377815349

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,changed

Author

Ilya Gutkovskiy, Jun 15 2018

Status

approved