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 = 8, r = 9.
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..200
J-C. Aval, Multivariate Fuss-Catalan Numbers, arXiv:0711.0906 [math.CO], 2007.
J-C. Aval, Multivariate Fuss-Catalan Numbers, Discrete Math., 308 (2008), 4660-4669.
Thomas A. Dowling, Catalan Numbers Chapter 7
Wojciech Mlotkowski, Fuss-Catalan Numbers in Noncommutative Probability, Docum. Mathm. 15: 939-955.
Wikipedia, Fuss-Catalan number
FORMULA
G.f. satisfies: A(x) = {1 + x*A(x)^(p/r)}^r, where p = 8, r = 9.
From Peter Bala, Oct 16 2015: (Start)
O.g.f.: (1/x) * series reversion (x*C(-x)^9), 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)^(1/9) is the o.g.f. for A007556. (End)
D-finite with recurrence +7*n*(7*n+3)*(7*n+4)*(7*n+5)*(7*n+6)*(7*n+8)*(7*n+9)*a(n)-128*(2*n+1)*(4*n+1)*(4*n+3)*(8*n+1)*(8*n+3)*(8*n+5)*(8*n+7)*a(n-1) = 0. - R. J. Mathar, Feb 09 2020
E.g.f.: F([9/8, 5/4, 11/8, 3/2, 13/8, 7/4, 15/8], [1, 10/7, 11/7, 12/7, 13/7, 15/7, 16/7], 16777216*x/823543), where F is the generalized hypergeometric function. - Stefano Spezia, Feb 09 2020
MATHEMATICA
Table[9 Binomial[8 n + 9, n]/(8 n + 9), {n, 0, 40}] (* Vincenzo Librandi, Dec 26 2013 *)
PROG
(PARI) a(n) = 9*binomial(8*n+9, n)/(8*n+9);
(PARI) {a(n)=local(B=1); for(i=0, n, B=(1+x*B^(8/9))^9+x*O(x^n)); polcoeff(B, n)}
(Magma) [9*Binomial(8*n+9, n)/(8*n+9): n in [0..30]]; // Vincenzo Librandi, Dec 26 2013
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Tim Fulford, Dec 26 2013
STATUS
approved