A305547 Expansion of e.g.f. Product_{k>=1} (1 + (exp(x) - 1)^k/k!).

1, 1, 2, 8, 37, 182, 1039, 7149, 56382, 479220, 4280247, 40406984, 410453366, 4539623168, 54431372233, 695801259947, 9312538336475, 128985882874288, 1842668013046405, 27238267120063415, 419396473955088310, 6769168354222927254, 114837651830425810381, 2042782103293394499566

Offset

0,3

Comments

Stirling transform of A007837.

Links

Seiichi Manyama, Table of n, a(n) for n = 0..500

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)/((j!)^k*k)).

a(n) = Sum_{k=0..n} Stirling2(n,k)*A007837(k).

Maple

b:= proc(n) option remember; `if`(n=0, 1, add(add((-d)*(-d!)^(-k/d),
       d=numtheory[divisors](k))*(n-1)!/(n-k)!*b(n-k), k=1..n))
    end:
a:= n-> add(Stirling2(n, k)*b(k), k=0..n):
seq(a(n), n=0..25);  # Alois P. Heinz, Jun 15 2018

Mathematica

nmax = 23; CoefficientList[Series[Product[(1 + (Exp[x] - 1)^k/k!), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 23; CoefficientList[Series[Exp[Sum[Sum[(-1)^(k + 1) (Exp[x] - 1)^(j k)/((j!)^k 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, -# (-#!)^(-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, 23}]

Crossrefs

Cf. A007837, A140585, A305987.

Sequence in context: A224032 A046814 A361698 * A007857 A289541 A047729

Adjacent sequences: A305544 A305545 A305546 * A305548 A305549 A305550

Keyword

nonn

Author

Ilya Gutkovskiy, Jun 15 2018

Status

approved