OFFSET
0,2
COMMENTS
Fuss-Catalan sequence is a(n,p,r) = r*binomial(np+r,n)/(np+r), where p = 9, 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. Mathm. 15 (2010), 939-955.
Wikipedia, Fuss-Catalan number.
FORMULA
G.f. satisfies: A(x) = {1 + x*A(x)^(p/r)}^r, where p = 9, r = 7.
O.g.f. A(x) = 1/x * series reversion (x/C(x)^7), where C(x) is the o.g.f. for the Catalan numbers A000108. A(x)^(1/7) is the o.g.f. for A059967. - Peter Bala, Oct 14 2015
a(n) ~ 7 * 3^(18*n+13) / (2^(24*n+23) * n^(3/2) * sqrt(Pi)). - Amiram Eldar, Sep 12 2025
D-finite with recurrence 128*n*(8*n+3)*(4*n+3)*(8*n+1)*(2*n+1)*(8*n+7)*(4*n+1)*(8*n+5)*a(n) -81*(9*n+2)*(9*n+4)*(3*n+2)*(9*n-1)*(9*n+1)*(3*n+1)*(9*n+5)*(9*n-2)*a(n-1)=0. - R. J. Mathar, Mar 25 2026
MATHEMATICA
Table[7 Binomial[9 n + 7, n]/(9 n + 7), {n, 0, 40}] (* Vincenzo Librandi, Dec 27 2013 *)
PROG
(PARI) a(n) = 7*binomial(9*n+7, n)/(9*n+7);
(PARI) {a(n)=local(B=1); for(i=0, n, B=(1+x*B^(9/7))^7+x*O(x^n)); polcoeff(B, n)}
(Magma) [7*Binomial(9*n+7, n)/(9*n+7): n in [0..30]]; // Vincenzo Librandi, Dec 27 2013
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Tim Fulford, Dec 27 2013
STATUS
approved