A386415 - OEIS

A386415

G.f. A(x) satisfies A(x) = (1 + 9*x*A(x)^4)^(2/3).

1, 6, 135, 4140, 146475, 5629338, 228355281, 9622693080, 417122726490, 18480617374050, 833136935399208, 38094723501749460, 1762459398803643930, 82353342267057244950, 3880848811889775489300, 184228926273804535479216, 8801795826996054546077865, 422898288144162288398536860

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OFFSET

0,2

LINKS

Paolo Xausa, Table of n, a(n) for n = 0..500

Wikipedia, Fuss-Catalan number.

FORMULA

a(n) = 9^n * binomial((8*n+2)/3,n)/(4*n+1).

G.f.: B(x)^2, where B(x) is the g.f. of A386416.

D-finite with recurrence 5*n*(n-1)*(n-2)*(5*n-4)*(5*n+2)*(5*n-7)*(5*n-1)*a(n) - 3456*(4*n-11)*(8*n-19)*(8*n-13)*(4*n-5)*(8*n-7)*(2*n-1)*(8*n-1)*a(n-3) = 0. - R. J. Mathar, Jul 30 2025

a(n) ~ 2^(8*n+1) * 3^n / (5^(5*n/3+7/6) * n^(3/2) * sqrt(Pi)). - Amiram Eldar, Nov 21 2025

MAPLE

A386415 := proc(n)

9^n*binomial((8*n+2)/3, n)/(4*n+1) ;

end proc:

seq(A386415(n), n=0..80) ; # R. J. Mathar, Jul 30 2025

MATHEMATICA

A386415[n_] := 9^n * Binomial[(8*n + 2)/3, n]/(4*n + 1);

Array[A386415, 20, 0] (* Paolo Xausa, Aug 01 2025 *)

PROG

(PARI) apr(n, p, r) = r*binomial(n*p+r, n)/(n*p+r);

a(n) = 9^n*apr(n, 8/3, 2/3);