OFFSET
0,2
COMMENTS
Fuss-Catalan sequence is a(n,p,r) = r*binomial(n*p + r,n)/(n*p + r), this is the case p = 8, r = 2.
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.
Gi-Sang Cheon, S.-T. Jin, and L. W. Shapiro, A combinatorial equivalence relation for formal power series, Linear Algebra and its Applications, Volume 491, 15 February 2016, Pages 123-137.
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]
Clemens Heuberger, Sarah J. Selkirk, and Stephan Wagner, Enumeration of Generalized Dyck Paths Based on the Height of Down-Steps Modulo k, arXiv:2204.14023 [math.CO], 2022.
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 = 8, r = 2.
a(n) = 2*binomial(8n+1,n-1)/n for n>0, a(0)=1. - Bruno Berselli, Jan 19 2014
A(x^3) = 1/x * series reversion (x/C(x^3)^2), where C(x) = (1 - sqrt(1 - 4*x))/(2*x) is the o.g.f. for the Catalan numbers A000108. A(x)^(1/2) is the o.g.f. for A007556. - Peter Bala, Oct 14 2015
a(n) ~ 2^(24*n+5) / (7^(7*n+5/2) * n^(3/2) * sqrt(Pi)). - Amiram Eldar, Sep 14 2025
D-finite with recurrence 7*n*(7*n-3)*(7*n+1)*(7*n-2)*(7*n+2)*(7*n-1)*(7*n-4)*a(n) -128*(8*n-5)*(4*n-1)*(8*n+1)*(2*n-1)*(8*n-1)*(4*n-3)*(8*n-3)*a(n-1)=0. - R. J. Mathar, Mar 18 2026
MATHEMATICA
Table[Binomial[8 n + 2, n]/(4 n + 1), {n, 0, 30}]
PROG
(PARI) a(n) = binomial(8*n+2, n)/(4*n+1);
(PARI) {a(n)=local(B=1); for(i=0, n, B=(1+x*B^4)^2+x*O(x^n)); polcoeff(B, n)}
(Magma) [Binomial(8*n+2, n)/(4*n+1): n in [0..30]];
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Tim Fulford, Dec 26 2013
STATUS
approved