A233738 2*binomial(5*n+10, n)/(n+2).

1, 10, 95, 920, 9135, 92752, 959595, 10084360, 107375730, 1156073100, 12565671261, 137702922560, 1519842008360, 16880051620320, 188519028884675, 2115822959020080, 23851913523156675, 269958280013904870, 3066451080298820830, 34946186787944832400

Offset: 0

Author

Tim Fulford, Dec 15 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=5, r=10.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• J-C. Aval, Multivariate Fuss-Catalan Numbers, arXiv:0711.0906 [math.CO], 2007.
• J-C. Aval, Multivariate Fuss-Catalan Numbers, 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, A004344, A118969, A118971, A143546, A233668, A233669, A233736, A233737.

Programs

Magma

[2*Binomial(5*n+10, n)/(n+2): n in [0..30]]; // Vincenzo Librandi, Dec 16 2013

Maple

A233738:=n->2*binomial(5*n+10,n)/(n+2): seq(A233738(n), n=0..30); # Wesley Ivan Hurt, Sep 07 2014

Mathematica

Table[2 Binomial[5 n + 10, n]/(n + 2), {n, 0, 40}] (* Vincenzo Librandi, Dec 16 2013 *)

PARI

a(n) = 2*binomial(5*n+10,n)/(n+2);

PARI

{a(n)=local(B=1); for(i=0, n, B=(1+x*B^(1/2))^10+x*O(x^n)); polcoeff(B, n)}

Formula

G.f. satisfies: B(x) = {1 + x*B(x)^(p/r)}^r, where p=5, r=10.

a(n) = 2*A004344(n)/(n+2). - Wesley Ivan Hurt, Sep 07 2014

G.f.: hypergeom([2, 11/5, 12/5, 13/5, 14/5], [11/4, 3, 13/4, 7/2], (3125/256)*x). - Robert Israel, Sep 07 2014

D-finite with recurrence 8*(2*n+5)*(4*n+7)*(n+2)*(4*n+9)*a(n) -(n+1)*(13877*n^3+45630*n^2+46579*n+14034)*a(n-1) +210*(5*n+1)*(5*n+2)*(5*n+3)*(5*n+4)*a(n-2)=0. - R. J. Mathar, Nov 22 2024

D-finite with recurrence 8*n*(2*n+5)*(4*n+7)*(n+2)*(4*n+9)*a(n) -5*(5*n+6)*(5*n+7)*(5*n+8)*(5*n+9)*(n+1)*a(n-1)=0. - R. J. Mathar, Nov 22 2024