OFFSET
0,2
COMMENTS
Fuss-Catalan sequence is a(n,p,r) = r*binomial(n*p + r,n)/(n*p + r), where p = 9, r = 10.
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 = 9, r = 10.
From _Peter Bala, Oct 16 2015: (Start)
O.g.f. A(x) = 1/x * series reversion (x*C(-x)^10), 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)^(1/10) is the o.g.f. for A062994. (End)
D-finite with recurrence: 128*n*(8*n+3)*(4*n+3)*(8*n+9)*(2*n+1)*(8*n+7)*(4*n+5)*(8*n+5)*a(n) -81*(9*n+2)*(9*n+4)*(3*n+2)*(9*n+8)*(9*n+1)*(3*n+1)*(9*n+5)*(9*n+7)*a(n-1)=0. - R. J. Mathar, Feb 21 2020
MATHEMATICA
Table[10 Binomial[9 n + 10, n]/(9 n + 10), {n, 0, 30}]
PROG
(PARI) a(n) = 10*binomial(9*n+10, n)/(9*n+10);
(PARI) {a(n)=local(B=1); for(i=0, n, B=(1+x*B^(9/10))^10+x*O(x^n)); polcoeff(B, n)}
(Magma) [10*Binomial(9*n+10, n)/(9*n+10): n in [0..30]];
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Tim Fulford, Dec 28 2013
STATUS
approved