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Expansion of Sum_{k>=0} k!!*x^k/Product_{j=1..k} (1 - j*x).
2

%I #7 Aug 04 2021 09:55:28

%S 1,1,3,10,41,201,1126,7043,48603,366298,2987189,26163501,244654150,

%T 2430411335,25539609327,282834656434,3290175964577,40089424302657,

%U 510340938343270,6772086558823547,93481666812344979,1339885322519303434,19907413622297965373,306126204811557339045

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

%C Stirling transform of A006882.

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

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

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

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

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

%p b:= proc(n, m) option remember;

%p `if`(n=0, doublefactorial(m), m*b(n-1, m)+b(n-1, m+1))

%p end:

%p a:= n-> b(n, 0):

%p seq(a(n), n=0..23); # _Alois P. Heinz_, Aug 04 2021

%t nmax = 23; CoefficientList[Series[Sum[k!! x^k/Product[1 - j x, {j, 1, k}], {k, 0, nmax}], {x, 0, nmax}], x]

%t 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]!

%t Table[Sum[StirlingS2[n, k] k!!, {k, 0, n}], {n, 0, 23}]

%Y Cf. A000670, A004123, A006882, A305404.

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, May 31 2018