A233827 a(n) = 8*binomial(6*n+8,n)/(6*n+8).

1, 8, 76, 800, 8990, 105672, 1283464, 15981504, 202927725, 2617624680, 34206162848, 451872681728, 6024664312030, 80964348872400, 1095590286231120, 14915165412813184, 204140673966231870, 2807362363541687280, 38772186055550141700

Offset: 0

Author

Tim Fulford, Dec 16 2013

Keywords

• nonn

Comments

Fuss-Catalan sequence is a(n,p,r) = r*binomial(np+r,n)/(np+r), this is the case p=6, r=8.

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, A002295, A212071, A212072, A212073, A130564, A233743, A233829, A233830.

Programs

Magma

[8*Binomial(6*n+8, n)/(6*n+8): n in [0..30]];

Mathematica

Table[8 Binomial[6 n + 8, n]/(6 n + 8), {n, 0, 30}]

PARI

a(n) = 8*binomial(6*n+8,n)/(6*n+8);

PARI

{a(n)=local(B=1); for(i=0, n, B=(1+x*B^(6/8))^8+x*O(x^n)); polcoeff(B, n)}

Formula

G.f. satisfies: B(x) = {1 + x*B(x)^(p/r)}^r, where p=6, r=8.

From Ilya Gutkovskiy, Sep 14 2018: (Start)

E.g.f.: 5F5(4/3,3/2,5/3,11/6,13/6; 1,9/5,11/5,12/5,13/5; 46656*x/3125).

a(n) ~ 3^(6*n+15/2)*4^(3*n+5)/(sqrt(Pi)*5^(5*n+17/2)*n^(3/2)). (End)

D-finite with recurrence 5*n*(5*n+6)*(5*n+7)*(5*n+8)*(5*n+4)*a(n) -72*(6*n+5)*(3*n+2)*(2*n+1)*(3*n+1)*(6*n+7)*a(n-1)=0. - R. J. Mathar, Nov 22 2024