A234506 - OEIS

%I #27 Sep 08 2022 08:46:06

%S 1,3,30,406,6327,107019,1909908,35399520,674842149,13147742322,

%T 260626484118,5239783981320,106585537781775,2189670831627678,

%U 45366284782209600,946815917066740800,19887218367823853937,420076689292591271325,8917736795123409615060,190161017612160607167948,4071301730663135449185705

%N a(n) = binomial(9*n+3, n)/(3*n+1).

%C Fuss-Catalan sequence is a(n,p,r) = r*binomial(n*p + r, n)/(n*p + r), where p=9, r=3.

%H Vincenzo Librandi, <a href="/A234506/b234506.txt">Table of n, a(n) for n = 0..200</a>

%H J-C. Aval, <a href="http://arxiv.org/pdf/0711.0906v1.pdf">Multivariate Fuss-Catalan Numbers</a>, arXiv:0711.0906v1, Discrete Math., 308 (2008), 4660-4669.

%H Thomas A. Dowling, <a href="http://www.mhhe.com/math/advmath/rosen/r5/instructor/applications/ch07.pdf">Catalan Numbers Chapter 7</a>

%H Wojciech Mlotkowski, <a href="http://www.math.uiuc.edu/documenta/vol-15/28.pdf">Fuss-Catalan Numbers in Noncommutative Probability</a>, Docum. Mathm. 15: 939-955.

%F G.f. satisfies: B(x) = {1 + x*B(x)^(p/r)}^r, where p=9, r=3.

%t Table[Binomial[9n+3, n]/(3n+1), {n, 0, 30}]

%o (PARI) a(n) = binomial(9*n+3,n)/(3*n+1);

%o (PARI) {a(n)=local(B=1); for(i=0, n, B=(1+x*B^3)^3+x*O(x^n)); polcoeff(B, n)}

%o (Magma) [Binomial(9*n+3, n)/(3*n+1): n in [0..30]];

%o (Sage) [binomial(9*n+3, n)/(3*n+1) for n in (0..30)] # _G. C. Greubel_, Feb 09 2021

%Y Cf. A000108, A143554, A232265, A234505, A234507, A234508, A234509, A234510, A234513.

%K nonn

%O 0,2

%A _Tim Fulford_, Dec 27 2013