OFFSET
0,2
COMMENTS
Fuss-Catalan sequence is a(n,p,r) = r*binomial(np+r,n)/(np+r), this is the case p = 7, r = 5.
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..200
Jean-Christophe Aval, Multivariate Fuss-Catalan Numbers, Discrete Math., Vol. 308, No. 20 (2008), 4660-4669; arXiv preprint, arXiv:0711.0906 [math.CO], 2007.
Thomas A. Dowling, Catalan Numbers, Chapter 7 of Applications of discrete mathematics, John G. Michaels and Kenneth H. Rosen (eds.), McGraw-Hill, New York, 1991. [Wayback Machine link]
Wojciech Mlotkowski, Fuss-Catalan Numbers in Noncommutative Probability, Docum. Math. 15 (2010), 939-955.
Wikipedia, Fuss-Catalan number.
FORMULA
G.f. satisfies: A(x) = {1 + x*A(x)^(p/r)}^r, where p = 7, r = 5.
O.g.f.: A(x) = 1/x * series reversion (x/C(x)^5), where C(x) is the o.g.f. for the Catalan numbers A000108. A(x)^(1/5) is the o.g.f. for A002296. - Peter Bala, Oct 14 2015
a(n) ~ 5 * 7^(7*n+9/2) / (3^(6*n+11/2) * 64^(n+1) * n^(3/2) * sqrt(Pi)). - Amiram Eldar, Sep 13 2025
D-finite with recurrence 72*n*(6*n+5)*(3*n+2)*(2*n+1)*(3*n+1)*(6*n+1)*a(n) -7*(7*n+4)*(7*n+1)*(7*n-2)*(7*n+2)*(7*n-1)*(7*n+3)*a(n-1)=0. - R. J. Mathar, Mar 18 2026
MATHEMATICA
Table[5 Binomial[7 n + 5, n]/(7 n + 5), {n, 0, 30}]
PROG
(PARI) a(n) = 5*binomial(7*n+5, n)/(7*n+5);
(PARI) {a(n)=local(B=1); for(i=0, n, B=(1+x*B^(7/5))^5+x*O(x^n)); polcoeff(B, n)}
(Magma) [5*Binomial(7*n+5, n)/(7*n+5): n in [0..30]];
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Tim Fulford, Dec 16 2013
STATUS
approved