%I #48 Sep 14 2025 02:28:51

%S 1,10,125,1760,26650,423752,6978510,117998400,2036685765,35738059500,

%T 635627275767,11433154297760,207621482341000,3801296492623560,

%U 70092637731997100,1300500163756675200,24262157874835233000,454847339247972377850,8564398318045559667475,161895214788423913972000

%N a(n) = 5*binomial(8*n+10,n)/(4*n+5).

%C Fuss-Catalan sequence is a(n,p,r) = r*binomial(np+r,n)/(np+r), this is the case p=8, r=10.

%H Vincenzo Librandi, <a href="/A230390/b230390.txt">Table of n, a(n) for n = 0..200</a>

%H Jean-Christophe Aval, <a href="https://doi.org/10.1016/j.disc.2007.08.100">Multivariate Fuss-Catalan Numbers</a>, Discrete Math., Vol. 308, No. 20 (2008), 4660-4669; <a href="https://arxiv.org/abs/0711.0906">arXiv preprint</a>, arXiv:0711.0906 [math.CO], 2007.

%H Thomas A. Dowling, <a href="https://web.archive.org/web/20170830003716/http://www.mhhe.com/math/advmath/rosen/r5/instructor/applications/ch07.pdf">Catalan Numbers</a>, Chapter 7 of Applications of discrete mathematics, John G. Michaels and Kenneth H. Rosen (eds.), McGraw-Hill, New York, 1991. [Wayback Machine link]

%H Wojciech Mlotkowski, <a href="https://doi.org/10.4171/dm/318">Fuss-Catalan Numbers in Noncommutative Probability</a>, Docum. Math. 15 (2010), 939-955.

%F G.f. satisfies: A(x) = (1 + x*A(x)^(p/r))^r, where p=8, r=10.

%F a(n) ~ 5 * 2^(24*n+29) / (7^(7*n+21/2) * n^(3/2) * sqrt(Pi)). - _Amiram Eldar_, Sep 14 2025

%t Table[5 Binomial[8 n + 10, n]/(4 n + 5), {n, 0, 30}]

%o (PARI) a(n) = 5*binomial(8*n+10,n)/(4*n+5);

%o (PARI) {a(n)=local(B=1); for(i=0, n, B=(1+x*B^(4/5))^10+x*O(x^n)); polcoeff(B, n)}

%o (Magma) [5*Binomial(8*n+10, n)/(4*n+5): n in [0..30]];

%Y Cf. A000108, A007556, A234461, A234462, A234463, A234464, A234465, A234466, A234467.

%K nonn

%O 0,2

%A _Tim Fulford_, Dec 28 2013