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

Offset: 0

Author

Seiichi Manyama, Jul 21 2025

Keywords

• nonn
• easy

Links

• Paolo Xausa, Table of n, a(n) for n = 0..500
• Wikipedia, Fuss-Catalan number

Crossrefs

Cf. A004989, A377269, A386413, A386414.

Cf. A078532, A386416.

Programs

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 *)

PARI

apr(n, p, r) = r*binomial(n*p+r, n)/(n*p+r);
a(n) = 9^n*apr(n, 8/3, 2/3);

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