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 = 3.
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
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. Mathm. 15: 939-955.
Wikipedia, Fuss-Catalan number
FORMULA
G.f. satisfies: A(x) = {1 + x*A(x)^(p/r)}^r, where p = 8, r = 3.
A(x^2) = 1/x * series reversion (x/C(x^2)^3), where C(x) = (1 - sqrt(1 - 4*x))/(2*x) is the o.g.f. for the Catalan numbers A000108. A(x)^(1/3) is the o.g.f. for A007556. - Peter Bala, Oct 14 2015
MATHEMATICA
Table[3 Binomial[8 n + 3, n]/(8 n + 3), {n, 0, 40}] (* Vincenzo Librandi, Dec 26 2013 *)
PROG
(PARI) a(n) = 3/(8*n+3)*binomial(8*n+3, n);
(PARI) {a(n)=local(B=1); for(i=0, n, B=(1+x*B^(8/3))^3+x*O(x^n)); polcoeff(B, n)}
(Magma) [3*Binomial(8*n+3, n)/(8*n+3): n in [0..30]]; // Vincenzo Librandi, Dec 26 2013
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Tim Fulford, Dec 26 2013
STATUS
approved