A305405 Expansion of Sum_{k>=0} k!!*x^k/Product_{j=1..k} (1 - j*x).

1, 1, 3, 10, 41, 201, 1126, 7043, 48603, 366298, 2987189, 26163501, 244654150, 2430411335, 25539609327, 282834656434, 3290175964577, 40089424302657, 510340938343270, 6772086558823547, 93481666812344979, 1339885322519303434, 19907413622297965373, 306126204811557339045

Offset: 0

Author

Ilya Gutkovskiy, May 31 2018

Keywords

• nonn

Comments

Stirling transform of A006882.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..517
• N. J. A. Sloane, Transforms
• Eric Weisstein's World of Mathematics, Stirling Transform

Crossrefs

Cf. A000670, A004123, A006882, A305404.

Programs

Maple

b:= proc(n, m) option remember;
     `if`(n=0, doublefactorial(m), m*b(n-1, m)+b(n-1, m+1))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..23);  # Alois P. Heinz, Aug 04 2021

Mathematica

nmax = 23; CoefficientList[Series[Sum[k!! x^k/Product[1 - j x, {j, 1, k}], {k, 0, nmax}], {x, 0, nmax}], x]
nmax = 23; CoefficientList[Series[1 + Exp[(E^x - 1)^2/2] (Exp[x] - 1) (1 + Sqrt[Pi/2] Erf[(Exp[x] - 1)/Sqrt[2]]), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[StirlingS2[n, k] k!!, {k, 0, n}], {n, 0, 23}]

Formula

E.g.f.: 1 + exp((exp(x) - 1)^2/2)*(exp(x) - 1)*(1 + sqrt(Pi/2)*erf((exp(x) - 1)/sqrt(2))).

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