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 = 6, r = 7.
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 = 6, r = 7.
From Peter Bala, Oct 16 2015: (Start)
O.g.f.: A(x) = (1/x) * series reversion (x*C(-x)^7), where C(x) = (1 - sqrt(1 - 4*x))/(2*x) is the o.g.f. for the Catalan numbers A000108. See cross-references for other Fuss-Catalan sequences with o.g.f. (1/x) * series reversion (x*C(-x)^k), k = 3 through 11.
A(x)^(1/7) is the o.g.f. for A002295. (End)
a(n) ~ 7 * 2^(6*n+6) * 3^(6*n+13/2) / (5^(5*n+15/2) * n^(3/2) * sqrt(Pi)). - Amiram Eldar, Sep 16 2025
MATHEMATICA
Table[7 Binomial[6 n + 7, n]/(6 n + 7), {n, 0, 40}] (* Vincenzo Librandi, Dec 16 2013 *)
PROG
(PARI) a(n) = 7*binomial(6*n+7, n)/(6*n+7);
(PARI) {a(n)=local(B=1); for(i=0, n, B=(1+x*B^(6/7))^7+x*O(x^n)); polcoeff(B, n)}
(Magma) [7*Binomial(6*n+7, n)/(6*n+7): n in [0..30]]; // Vincenzo Librandi, Dec 16 2013
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Tim Fulford, Dec 15 2013
EXTENSIONS
More terms from Vincenzo Librandi, Dec 16 2013
STATUS
approved