A386414 G.f. A(x) satisfies A(x) = (1 + 9*x*A(x)^3)^(2/3).

1, 6, 99, 2142, 52785, 1404702, 39331656, 1141839504, 34057559052, 1037385419400, 32133013365915, 1009060082062110, 32050934711814915, 1027914968037080970, 33240367148212098900, 1082645830435810233960, 35483717092533680418039, 1169426742892003447650666

Offset

0,2

Links

Paolo Xausa, Table of n, a(n) for n = 0..600

Wikipedia, Fuss-Catalan number.

Formula

a(n) = 9^n * binomial((6*n+2)/3,n)/(3*n+1).

G.f.: B(x)^2, where B(x) is the g.f. of A008931.

From R. J. Mathar, Jul 30 2025: (Start)

D-finite with recurrence n*(3*n+2)*a(n) - 6*(6*n-1)*(3*n-2)*a(n-1) = 0.

G.f.: 2F1(1/3,5/6 ; 5/3 ; 36*x). (End)

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

Mathematica

A386414[n_] := 9^n*Binomial[(6*n + 2)/3, n]/(3*n + 1);
Array[A386414, 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, 2, 2/3);

Crossrefs

Cf. A004989, A377269, A386413, A386415.

Cf. A004987, A008931.

Sequence in context: A244251 A187522 A224615 * A138913 A360336 A192667

Adjacent sequences: A386411 A386412 A386413 * A386415 A386416 A386417

Keyword

nonn,easy

Author

Seiichi Manyama, Jul 21 2025

Status

approved