%I #18 Sep 10 2023 08:39:12

%S 1,1,5,42,498,7644,144156,3225648,83536008,2457701928,80970232104,

%T 2953056534768,118112744060208,5140622709134496,241863782829704928,

%U 12232551538417012992,661818290353375962240,38140594162828447248000,2332567001993176540206720,150880256846462633823648000

%N Expansion of e.g.f. 1 / (1 + 3 * log(1 - x))^(1/3).

%H Seiichi Manyama, <a href="/A347015/b347015.txt">Table of n, a(n) for n = 0..372</a>

%F a(n) = Sum_{k=0..n} |Stirling1(n,k)| * A007559(k).

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

%F a(0) = 1; a(n) = Sum_{k=1..n} (3 - 2*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, (3*n-2)*g(n-1)) end:

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

%p seq(a(n), n=0..19); # _Alois P. Heinz_, Aug 10 2021

%t nmax = 19; CoefficientList[Series[1/(1 + 3 Log[1 - x])^(1/3), {x, 0, nmax}], x] Range[0, nmax]!

%t Table[Sum[Abs[StirlingS1[n, k]] 3^k Pochhammer[1/3, k], {k, 0, n}], {n, 0, 19}]

%Y Cf. A007559, A007840, A346978, A346982, A347016.

%Y Cf. A354263.

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, Aug 10 2021