A233737 a(n) = 9*binomial(5*n+9, n)/(5*n+9).

1, 9, 81, 759, 7371, 73656, 752913, 7838298, 82832706, 886322710, 9583986555, 104568156819, 1149793519368, 12728471356944, 141747219186705, 1586867219265060, 17848735288114995, 201607141031660871, 2285899896222757346, 26008027474874327190, 296840444852078282610, 3397721117411729991960

Offset: 0

Author

Tim Fulford, Dec 15 2013

Keywords

• nonn

Comments

Fuss-Catalan sequence is a(n,p,r) = r*binomial(np+r,n)/(np+r), this is the case p=5, r=9.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• J-C. Aval, Multivariate Fuss-Catalan Numbers, arXiv:0711.0906v1, Discrete Math., 308 (2008), 4660-4669.
• Thomas A. Dowling, Catalan Numbers Chapter 7
• Wojciech Mlotkowski, Fuss-Catalan Numbers in Noncommutative Probability, Docum. Mathm. 15: 939-955.

Crossrefs

Cf. A000108, A002294, A118969, A143546, A118971, A233668, A233669, A233736, A233738.

Programs

Magma

[9*Binomial(5*n+9,n)/(5*n+9): n in [0..30]];

Mathematica

Table[9 Binomial[5 n + 9, n]/(5 n + 9), {n, 0, 30}]

PARI

a(n) = 9*binomial(5*n+9,n)/(5*n+9);

PARI

{a(n)=local(B=1); for(i=0, n, B=(1+x*B^(5/9))^9+x*O(x^n)); polcoeff(B, n)}

Formula

G.f. satisfies: B(x) = {1 + x*B(x)^(p/r)}^r, here p=5, r=9.

From Ilya Gutkovskiy, Sep 14 2018: (Start)

E.g.f.: 5F5(9/5,2,11/5,12/5,13/5; 1,5/2,11/4,3,13/4; 3125*x/256).

a(n) ~ 9*5^(5*n+17/2)/(sqrt(Pi)*2^(8*n+39/2)*n^(3/2)). (End)