A386413 G.f. A(x) satisfies A(x) = (1 + 9*x*A(x)^2)^(2/3).

1, 6, 63, 792, 10935, 160056, 2438667, 38263752, 614014830, 10029572280, 166203389781, 2787232297680, 47213065271268, 806618756189736, 13883029872725475, 240491818267745760, 4189678646994012501, 73357895462268102840, 1290223574267814268290, 22784365638084466567800

Offset: 0

Author

Seiichi Manyama, Jul 21 2025

Keywords

• nonn
• easy

Links

• Paolo Xausa, Table of n, a(n) for n = 0..750
• Wikipedia, Fuss-Catalan number

Crossrefs

Cf. A004989, A377269, A386414, A386415.

Cf. A078532, A376636.

Programs

Mathematica

A386413[n_] := 9^n*Binomial[(4*n + 2)/3, n]/(2*n + 1);
Array[A386413, 25, 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, 4/3, 2/3);

Formula

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

G.f.: B(x)^2, where B(x) is the g.f. of A078532.

D-finite with recurrence n*(n-2)*(n+2)*a(n) -216*(2*n-5)*(4*n-7)*(4*n-1)*a(n-3)=0. - R. J. Mathar, Jul 30 2025