This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A305405 #8 Feb 16 2025 08:33:54
%S A305405 1,1,3,10,41,201,1126,7043,48603,366298,2987189,26163501,244654150,
%T A305405 2430411335,25539609327,282834656434,3290175964577,40089424302657,
%U A305405 510340938343270,6772086558823547,93481666812344979,1339885322519303434,19907413622297965373,306126204811557339045
%N A305405 Expansion of Sum_{k>=0} k!!*x^k/Product_{j=1..k} (1 - j*x).
%C A305405 Stirling transform of A006882.
%H A305405 Alois P. Heinz, <a href="/A305405/b305405.txt">Table of n, a(n) for n = 0..517</a>
%H A305405 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%H A305405 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/StirlingTransform.html">Stirling Transform</a>
%F A305405 E.g.f.: 1 + exp((exp(x) - 1)^2/2)*(exp(x) - 1)*(1 + sqrt(Pi/2)*erf((exp(x) - 1)/sqrt(2))).
%F A305405 a(n) = Sum_{k=0..n} Stirling2(n,k)*k!!.
%p A305405 b:= proc(n, m) option remember;
%p A305405 `if`(n=0, doublefactorial(m), m*b(n-1, m)+b(n-1, m+1))
%p A305405 end:
%p A305405 a:= n-> b(n, 0):
%p A305405 seq(a(n), n=0..23); # _Alois P. Heinz_, Aug 04 2021
%t A305405 nmax = 23; CoefficientList[Series[Sum[k!! x^k/Product[1 - j x, {j, 1, k}], {k, 0, nmax}], {x, 0, nmax}], x]
%t A305405 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 A305405 Table[Sum[StirlingS2[n, k] k!!, {k, 0, n}], {n, 0, 23}]
%Y A305405 Cf. A000670, A004123, A006882, A305404.
%K A305405 nonn
%O A305405 0,3
%A A305405 _Ilya Gutkovskiy_, May 31 2018