OFFSET
0,2
COMMENTS
Fuss-Catalan sequence is a(n,p,r) = r*binomial(np+r,n)/(np+r), where p = 10, r = 8.
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 = 10, r = 8.
O.g.f. A(x) = 1/x * series reversion (x/C(x)^8), where C(x) is the o.g.f. for the Catalan numbers A000108. A(x)^(1/8) is the o.g.f. for A059968. - Peter Bala, Oct 14 2015
a(n) ~ 2^(10*n+10) * 5^(10*n+15/2) / (3^(18*n+17) * n^(3/2) * sqrt(Pi)). - Amiram Eldar, Sep 15 2025
D-finite with recurrence 81*n*(9*n+2)*(9*n+4)*(3*n+2)*(9*n+8)*(9*n+1)*(3*n+1)*(9*n+5)*(9*n+7)*a(n) -800*(10*n+1)*(5*n+1)*(10*n+3)*(5*n+2)*(2*n+1)*(5*n+3)*(10*n+7)*(5*n-1)*(10*n-1)*a(n-1)=0. - R. J. Mathar, Mar 18 2026
MATHEMATICA
Table[4 Binomial[10 n + 8, n]/(5 n + 4), {n, 0, 30}]
PROG
(PARI) a(n) = 4*binomial(10*n+8, n)/(5*n+4);
(PARI) {a(n)=local(B=1); for(i=0, n, B=(1+x*B^(5/4))^8+x*O(x^n)); polcoeff(B, n)}
(Magma) [4*Binomial(10*n+8, n)/(5*n+4): n in [0..30]];
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Tim Fulford, Dec 28 2013
STATUS
approved