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=2.
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..200
J-C. Aval, Multivariate Fuss-Catalan Numbers, arXiv:0711.0906v1, 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
Jun Yan, Lattice paths enumerations weighted by ascent lengths, arXiv:2501.01152 [math.CO], 2025. See p. 7.
Sheng-liang Yang and Mei-yang Jiang, Pattern avoiding problems on the hybrid d-trees, J. Lanzhou Univ. Tech., (China, 2023) Vol. 49, No. 2, 144-150. (in Mandarin)
FORMULA
G.f. satisfies: B(x) = {1 + x*B(x)^(p/r)}^r, where p=7, r=2.
a(n) = 2*binomial(7n+1, n-1)/n for n>0, a(0)=1. [Bruno Berselli, Jan 19 2014]
From Ilya Gutkovskiy, Sep 14 2018: (Start)
E.g.f.: 6F6(2/7,3/7,4/7,5/7,6/7,8/7; 1/2,2/3,5/6,1,7/6,4/3; 823543*x/46656).
a(n) ~ 7^(7*n+3/2)/(sqrt(Pi)*3^(6*n+5/2)*4^(3*n+1)*n^(3/2)). (End)
MATHEMATICA
Table[2 Binomial[7 n + 2, n]/(7 n + 2), {n, 0, 30}]
PROG
(PARI) a(n) = 2*binomial(7*n+2, n)/(7*n+2);
(PARI) {a(n)=local(B=1); for(i=0, n, B=(1+x*B^(7/2))^2+x*O(x^n)); polcoeff(B, n)}
(Magma) [2*Binomial(7*n+2, n)/(7*n+2): n in [0..30]];
CROSSREFS
KEYWORD
nonn
AUTHOR
Tim Fulford, Dec 16 2013
STATUS
approved