A373517 Expansion of e.g.f. exp(x/(1 - x^3)^(1/3)).
1, 1, 1, 1, 9, 41, 121, 1401, 11761, 61489, 864081, 10597841, 81833401, 1350154521, 21715461769, 225232218121, 4267472824161, 84597818284001, 1111699778741281, 23801969674626849, 558853937533757161, 8943028907965939081, 213696639293901810201
Offset: 0
Programs
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Mathematica
nmax = 25; CoefficientList[Series[E^(x/(1 - x^3)^(1/3)), {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Sep 03 2025 *) -
PARI
a(n) = n!*sum(k=0, n\3, binomial(n/3-1, k)/(n-3*k)!);
Formula
a(n) = n! * Sum_{k=0..floor(n/3)} binomial(n/3-1,k)/(n-3*k)!.
a(n) == 1 mod 8.
From Vaclav Kotesovec, Sep 03 2025: (Start)
a(n) = (4*n^3 - 36*n^2 + 112*n - 119)*a(n-3) - 2*(n-6)*(n-5)*(n-4)*(n-3)*(3*n^2 - 27*n + 64)*a(n-6) + 4*(n-9)*(n-8)*(n-7)*(n-6)^3*(n-5)*(n-4)*(n-3)*a(n-9) - (n-12)*(n-11)*(n-10)*(n-9)^2*(n-8)*(n-7)*(n-6)^2*(n-5)*(n-4)*(n-3)*a(n-12).
a(n) ~ (1/2) * exp(4*n^(1/4)/3 - n) * n^(n - 3/8) * (1 - 35/(96*n^(1/4)) - 4367/(18432*sqrt(n)) + 1737829/(5308416*n^(3/4))). (End)