OFFSET
0,2
COMMENTS
Fuss-Catalan sequence is a(n,p,r) = r*binomial(n*p + r,n)/(n*p + r), where p = 10, r = 11.
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 = 10, r = 11.
From Peter Bala, Oct 16 2015: (Start)
O.g.f.: A(x) = (1/x) * series reversion (x*C(-x)^11), 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)^(1/11) is the o.g.f. for A059968. (End)
D-finite with recurrence: 81*n*(9*n+11)*(9*n+4)*(3*n+2)*(9*n+8)*(9*n+10)*(3*n+1)*(9*n+5)*(9*n+7)*a(n) - 800*(10*n+1)*(5*n+1)*(10*n+3)*(5*n+2)*(2*n+1)*(5*n+3)*(10*n+7)*(5*n+4)*(10*n+9)*a(n-1) = 0. - R. J. Mathar, Feb 21 2020
a(n) ~ 11 * 2^(10*n+10) * 5^(10*n+21/2) / (3^(18*n+23) * n^(3/2) * sqrt(Pi)). - Amiram Eldar, Sep 15 2025
MATHEMATICA
Table[11/(10 n + 11) Binomial[10 n + 11, n], {n, 0, 40}] (* Vincenzo Librandi, Jan 10 2014 *)
PROG
(PARI) a(n) = 11*binomial(10*n+11, n)/(10*n+11);
(PARI) {a(n)=local(B=1); for(i=0, n, B=(1+x*B^(10/11))^11+x*O(x^n)); polcoeff(B, n)}
(Magma) [11*Binomial(10*n+11, n)/(10*n+11) : n in [0..20]]; // Vincenzo Librandi, Jan 10 2014
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Tim Fulford, Oct 04 2013
EXTENSIONS
Corrected by Vincenzo Librandi, Jan 10 2014
STATUS
approved