OFFSET
0,2
COMMENTS
Fuss-Catalan sequence is a(n,p,r) = r*binomial(np+r,n)/(np+r), where p=10, r=9.
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]
Elżbieta Liszewska and Wojciech Młotkowski, Some relatives of the Catalan sequence, arXiv:1907.10725 [math.CO], 2019.
Wojciech Mlotkowski, Fuss-Catalan Numbers in Noncommutative Probability, Docum. Math. 15 (2010), 939-955.
J. Sawada, J. Sears, A. Trautrim, and A. Williams, Demystifying our Grandparent's De Bruijn Sequences with Concatenation Trees, arXiv:2308.12405 [math.CO], 2023-2024.
FORMULA
G.f. satisfies: A(x) = (1 + x*A(x)^(p/r))^r, where p=10, r=9.
From Wolfdieter Lang, Feb 06 2020: (Start)
G.f.: hypergeom([9, 10, ..., 18]/10, [10, 11, ..., 18]/9, (10^10/9^9)*x).
E.g.f.: hypergeom([9, 11, 12, ..., 18]/10, [10, 11, ..., 18]/9, (10^10/9^9) * x). Cf. _Ilya Gutkovsky_ in A118971. (End)
a(n) = binomial(10*n + 8 , n+1)/(9*n + 8) which is instance k = 9 of c(k, n+1) given in a comment in A130564. x*A(x), with the above given g.f. A(x), is the compositional inverse of y*(1 - y)^9, hence A(x)*(1 - x*A(x))^9 = 1. For another formula for A(x) involving the hypergeometric function 9F8 see the analog formula in A234513. - Wolfdieter Lang, Feb 15 2024
a(n) ~ 4^(5*n+4) * 5^(10*n+17/2) / (3^(18*n+17) * n^(3/2) * sqrt(Pi)). - Amiram Eldar, Sep 14 2025
MATHEMATICA
Table[9 Binomial[10 n + 9, n]/(10 n + 9), {n, 0, 30}]
PROG
(PARI) a(n) = 9*binomial(10*n+9, n)/(10*n+9);
(PARI) {a(n)=local(B=1); for(i=0, n, B=(1+x*B^(10/9))^9+x*O(x^n)); polcoeff(B, n)}
(Magma) [9*Binomial(10*n+9, n)/(10*n+9): n in [0..30]];
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Tim Fulford, Dec 28 2013
STATUS
approved