A233666 a(n) = 2*binomial(4*n + 8, n)/(n + 2).

1, 8, 60, 456, 3542, 28080, 226548, 1855040, 15380937, 128896456, 1090119316, 9292881360, 79769043900, 688915123680, 5981962494852, 52193342019456, 457367224685012, 4023551800087200, 35521420783728880, 314608026125871720, 2794654131668318430

Offset: 0

Author

Tim Fulford, Dec 14 2013

Keywords

• nonn
• easy

Comments

Fuss-Catalan sequence is a(n,p,r) = r*binomial(np+r,n)/(np+r), this is the case p=4, 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, A002293, A069271, A006632, A196678, A006633, A233658, A006634, A233667, A006635.

Programs

Magma

[2*Binomial(4*n+8,n)/(n+2): n in [0..30]]; // Vincenzo Librandi, Dec 14 2013

Mathematica

Table[2/(n + 2) Binomial[4 n + 8, n], {n, 0, 40}] (* Vincenzo Librandi, Dec 14 2013 *)

PARI

a(n) = 4*binomial(4*n+8,n)/(n+2);

PARI

{a(n)=local(B=1); for(i=0, n, B=(1+x*B^(1/2))^8+x*O(x^n)); polcoeff(B, n)}

Formula

G.f. satisfies: B(x) = {1 + x*B(x)^(p/r)}^r, where p=4, r=8.

From Ilya Gutkovskiy, Sep 14 2018: (Start)

E.g.f.: 4F4(2,9/4,5/2,11/4; 1,3,10/3,11/3; 256*x/27).

a(n) ~ 2^(8*n+35/2)/(sqrt(Pi)*3^(3*n+17/2)*n^(3/2)). (End)

D-finite with recurrence 3*(3*n+7)*(n+2)*(3*n+8)*a(n) -2*(n+1)*(317*n^2+954*n+709)*a(n-1) +112*(4*n+1)*(2*n+1)*(4*n+3)*a(n-2)=0. - R. J. Mathar, Nov 22 2024