editing

approved

Alois P. Heinz, <a href="/A305405/b305405.txt">Table of n, a(n) for n = 0..517</a>

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

approved

editing

proposed

approved

editing

proposed

allocated for Ilya Gutkovskiy

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

0,3

Stirling transform of A006882.

N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>

Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/StirlingTransform.html">Stirling Transform</a>

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!!.

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}]

Cf. A000670, A004123, A006882, A305404.

allocated

nonn

Ilya Gutkovskiy, May 31 2018

approved

editing

allocated for Ilya Gutkovskiy

allocated

approved