OFFSET
0,3
COMMENTS
Fuss-Catalan sequence is a(n,p,r) = r*binomial(np+r,n)/(np+r), this is the case p=11, r=1. Interesting property when r=1, a(n+1,p,1) = a(n,p,p) for n>=0.
This is also instance k = 10 of the generalized Catalan family {C(k, n)}_{n>=0} given in a comment of A130564. - Wolfdieter Lang, Feb 05 2024
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..200
J-C. Aval, Multivariate Fuss-Catalan Numbers, arXiv:0711.0906 [math.CO], 2008.
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.
J.-C. Novelli, J.-Y. Thibon, Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions, arXiv preprint arXiv:1403.5962, 2014
Wikipedia, Fuss-Catalan number
FORMULA
G.f. satisfies: B(x) = {1 + x*B(x)^(p/r)}^r, with p=11, r=1.
From Robert FERREOL, Apr 01 2015: (Start)
a(n) = binomial(11*n,n)/(10*n+1) = A062993(n+9, 9).
a(0) = 1; a(n) = Sum_{i1+i2+..i11=n-1} a(i1)*a(i2)*...*a(i11) for n>=1.
(End)
O.g.f.: hypergeometric([1,...,10]/11,[2,...,9,11]/10,(11^11/10^10)*x). For the e.g.f. put an extra 1 = 10/10 into the second part. - Wolfdieter Lang, Feb 05 2024
a(n) ~ (11^11/10^10)^n*sqrt(11/(2*Pi*(10*n)^3)). - Robert A. Russell, Jul 15 2024
MAPLE
seq(binomial(11*k+1, k)/(11*k+1), k=0..30); # Robert FERREOL, Apr 01 2015
n:=30:G:=series(RootOf(g = 1+x*g^11, g), x=0, n+1):seq(coeff(G, x, k), k=0..n); # Robert FERREOL, Apr 01 2015
MATHEMATICA
Table[Binomial[11 n + 1, n]/(11 n + 1), {n, 0, 30}] (* Vincenzo Librandi, Jan 01 2014 *)
PROG
(PARI) a(n) = binomial(11*n+1, n)/(11*n+1);
(PARI) {a(n)=local(B=1); for(i=0, n, B=(1+x*B^(11))+x*O(x^n)); polcoeff(B, n)}
(Magma) [Binomial(11*n+1, n)/(11*n+1): n in [0..30]]; // Vincenzo Librandi, Jan 01 2014
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Tim Fulford, Jan 01 2014
STATUS
approved