A373518 Expansion of e.g.f. exp(x/(1 - x^3)^(2/3)).
1, 1, 1, 1, 17, 81, 241, 3361, 32481, 183457, 2534561, 36903681, 325995121, 4808334961, 90981786897, 1126128625441, 18354227120321, 415821040873281, 6714588707173441, 122710186163310337, 3174234862391072721, 63597591858999638161, 1308604168710672673841
Offset: 0
Programs
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Mathematica
nmax = 25; CoefficientList[Series[E^(x/(1 - x^3)^(2/3)), {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Sep 03 2025 *) -
PARI
a(n) = n!*sum(k=0, n\3, binomial(2*n/3-k-1, k)/(n-3*k)!);
Formula
a(n) = n! * Sum_{k=0..floor(n/3)} binomial(2*n/3-k-1,k)/(n-3*k)!.
a(n) == 1 mod 16.
From Vaclav Kotesovec, Sep 03 2025: (Start)
Recurrence: (n-9)*(2*n - 21)*a(n) = (2*n - 21)*(5*n^4 - 90*n^3 + 515*n^2 - 1229*n + 1071)*a(n-3) - (n-6)*(n-5)*(n-4)*(n-3)*(20*n^4 - 570*n^3 + 5590*n^2 - 21846*n + 30303)*a(n-6) + (n-9)*(n-8)*(n-7)*(n-6)*(n-5)*(n-4)*(n-3)*(20*n^4 - 630*n^3 + 6940*n^2 - 31104*n + 49113)*a(n-9) - (n-11)*(n-10)*(n-9)*(n-8)*(n-7)*(n-6)*(n-5)*(n-4)*(n-3)*(2*n - 9)*(5*n^4 - 210*n^3 + 3215*n^2 - 20971*n + 47886)*a(n-12) + (n-15)*(n-14)^2*(n - 13)^2*(n-12)*(n-11)*(n-10)*(n-9)*(n-8)*(n-7)*(n-6)^2*(n-5)*(n-4)*(n-3)*(2*n - 9)*a(n-15).
a(n) ~ 2^(3/10) * 5^(-1/2) * exp(5*2^(-2/5)*n^(2/5)/3 - n) * n^(n - 3/10).
(End)