A386416 G.f. A(x) satisfies A(x) = (1 + 9*x*A(x)^8)^(1/3).

1, 3, 63, 1881, 65610, 2499336, 100777122, 4228144596, 182674383705, 8072369224920, 363154406671485, 16576444298006658, 765806677899249168, 35739548618003938440, 1682429522012566325460, 79793991407758199002740, 3809208342822290233767522, 182890356905449116974950200

Offset: 0

Author

Seiichi Manyama, Jul 21 2025

Keywords

• nonn
• easy

Links

• Paolo Xausa, Table of n, a(n) for n = 0..550
• Wikipedia, Fuss-Catalan number

Crossrefs

Cf. A004990, A377268, A376636, A004987, A078532, A245114, A008931, A376282.

Cf. A386415.

Programs

Mathematica

A386416[n_] := 9^n*Binomial[(8*n + 1)/3, n]/(8*n + 1);
Array[A386416, 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, 1/3);

Formula

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

G.f. A(x) satisfies A(x) = 1/A(-x*A(x)^13).

D-finite with recurrence 5*n*(n-1)*(n-2)*(5*n-8)*(5*n-11)*(5*n+1)*(5*n-2)*a(n) -3456*(8*n-11)*(8*n-5)*(4*n-1)*(8*n-23)*(2*n-5)*(8*n-17)*(4*n-7)*a(n-3)=0. - R. J. Mathar, Jul 30 2025