A305987 Expansion of e.g.f. Product_{k>=1} (1 + (exp(x) - 1)^k/k).

1, 1, 2, 9, 52, 355, 2976, 29897, 343988, 4423503, 63088600, 992691205, 17095554444, 319404545291, 6427307831504, 138546745515393, 3185841858310180, 77866726065935239, 2016161715005701128, 55127056896177521981, 1587073087715010466556, 47982707153606476112067, 1519931218769637781731712

Offset

0,3

Comments

Stirling transform of A007838.

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} (-1)^(k+1)*(exp(x) - 1)^(j*k)/(k*j^k)).

a(n) = Sum_{k=0..n} Stirling2(n,k)*A007838(k).

a(n) ~ exp(-gamma) * n! / (2 * log(2)^(n+1)), where 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)/j!*
      b(n-i*j, i-1)*(i-1)!^j, j=0..min(1, 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 = 22; CoefficientList[Series[Product[(1 + (Exp[x] - 1)^k/k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 22; CoefficientList[Series[Exp[Sum[Sum[(-1)^(k + 1) (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, 22}]

Crossrefs

Cf. A007838, A305547, A305550, A305986.

Sequence in context: A003584 A301928 A069271 * A355789 A231494 A006152

Adjacent sequences: A305984 A305985 A305986 * A305988 A305989 A305990

Keyword

nonn

Author

Ilya Gutkovskiy, Jun 15 2018

Status

approved