%I #15 Sep 10 2023 08:39:08

%S 1,1,4,26,234,2694,37812,626352,11962164,258787812,6255195168,

%T 167072685240,4886611129320,155335056242040,5332298685827760,

%U 196590247328769120,7747254471910795920,324986515253994589200,14458392906960271354560,679977065168639138610720

%N Expansion of e.g.f. 1 / sqrt(1 + 2 * log(1 - x)).

%H Seiichi Manyama, <a href="/A346978/b346978.txt">Table of n, a(n) for n = 0..390</a>

%F a(n) = Sum_{k=0..n} |Stirling1(n,k)| * (2*k-1)!!.

%F a(n) ~ n^n / (exp(n/2) * (exp(1/2) - 1)^(n + 1/2)). - _Vaclav Kotesovec_, Aug 09 2021

%F a(0) = 1; a(n) = Sum_{k=1..n} (2 - k/n) * (k-1)! * binomial(n,k) * a(n-k). - _Seiichi Manyama_, Sep 09 2023

%t nmax = 19; CoefficientList[Series[1/Sqrt[1 + 2 Log[1 - x]], {x, 0, nmax}], x] Range[0, nmax]!

%t Table[Sum[Abs[StirlingS1[n, k]] (2 k - 1)!!, {k, 0, n}], {n, 0, 19}]

%Y Cf. A001147, A007840, A088500, A305404, A320343.

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, Aug 09 2021