%I #16 Sep 10 2023 08:39:17

%S 1,1,6,62,916,17644,419360,11859840,388965600,14514046560,

%T 607165485120,28143329181120,1431690475207680,79302863940387840,

%U 4751108622148907520,306118435580577146880,21107196651940518551040,1550773243761690603179520,120947288498720390755353600

%N Expansion of e.g.f. 1 / (1 + 4 * log(1 - x))^(1/4).

%H Seiichi Manyama, <a href="/A347016/b347016.txt">Table of n, a(n) for n = 0..360</a>

%F a(n) = Sum_{k=0..n} |Stirling1(n,k)| * A007696(k).

%F a(n) ~ n! * exp(n/4) / (Gamma(1/4) * 2^(1/2) * n^(3/4) * (exp(1/4) - 1)^(n + 1/4)). - _Vaclav Kotesovec_, Aug 14 2021

%F a(0) = 1; a(n) = Sum_{k=1..n} (4 - 3*k/n) * (k-1)! * binomial(n,k) * a(n-k). - _Seiichi Manyama_, Sep 09 2023

%p g:= proc(n) option remember; `if`(n<2, 1, (4*n-3)*g(n-1)) end:

%p a:= n-> add(abs(Stirling1(n, k))*g(k), k=0..n):

%p seq(a(n), n=0..18); # _Alois P. Heinz_, Aug 10 2021

%t nmax = 18; CoefficientList[Series[1/(1 + 4 Log[1 - x])^(1/4), {x, 0, nmax}], x] Range[0, nmax]!

%t Table[Sum[Abs[StirlingS1[n, k]] 4^k Pochhammer[1/4, k], {k, 0, n}], {n, 0, 18}]

%Y Cf. A007696, A007840, A346978, A346983, A347015.

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, Aug 10 2021