A347015 Expansion of e.g.f. 1 / (1 + 3 * log(1 - x))^(1/3).

1, 1, 5, 42, 498, 7644, 144156, 3225648, 83536008, 2457701928, 80970232104, 2953056534768, 118112744060208, 5140622709134496, 241863782829704928, 12232551538417012992, 661818290353375962240, 38140594162828447248000, 2332567001993176540206720, 150880256846462633823648000

Offset: 0

Author

Ilya Gutkovskiy, Aug 10 2021

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..372

Crossrefs

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

Cf. A354263.

Programs

Maple

g:= proc(n) option remember; `if`(n<2, 1, (3*n-2)*g(n-1)) end:
a:= n-> add(abs(Stirling1(n, k))*g(k), k=0..n):
seq(a(n), n=0..19);  # Alois P. Heinz, Aug 10 2021

Mathematica

nmax = 19; CoefficientList[Series[1/(1 + 3 Log[1 - x])^(1/3), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[Abs[StirlingS1[n, k]] 3^k Pochhammer[1/3, k], {k, 0, n}], {n, 0, 19}]

Formula

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

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

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