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a(n) = 2*binomial(9*n+2,n)/(9*n+2).
9

%I #31 May 16 2023 12:21:37

%S 1,2,19,252,3885,65274,1159587,21421248,407337153,7920326700,

%T 156753610013,3147328992080,63951322669065,1312575792628356,

%U 27172514322677625,566707337222428800,11896007334177739113,251142622845893276190,5328891499524964282170

%N a(n) = 2*binomial(9*n+2,n)/(9*n+2).

%C Fuss-Catalan sequence is a(n,p,r) = r*binomial(np+r,n)/(np+r), where p=9, r=2.

%H Vincenzo Librandi, <a href="/A234505/b234505.txt">Table of n, a(n) for n = 0..200</a>

%H J-C. Aval, <a href="https://arxiv.org/abs/0711.0906">Multivariate Fuss-Catalan Numbers</a>, arXiv:0711.0906 [math.CO], 2007; 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="https://www.emis.de/journals/DMJDMV/vol-15/28.html">Fuss-Catalan Numbers in Noncommutative Probability</a>, Docum. Mathm. 15: 939-955.

%H Sheng-liang Yang and Mei-yang Jiang, <a href="https://journal.lut.edu.cn/EN/abstract/abstract528.shtml">Pattern avoiding problems on the hybrid d-trees</a>, J. Lanzhou Univ. Tech., (China, 2023) Vol. 49, No. 2, 144-150. (in Mandarin)

%F G.f. satisfies: B(x) = {1 + x*B(x)^(p/r)}^r, where p=9, r=2.

%F a(n) = 2*binomial(9n+1,n-1)/n for n>0, a(0)=1. [_Bruno Berselli_, Jan 19 2014]

%t Table[2 Binomial[9 n + 2, n]/(9 n + 2), {n, 0, 30}]

%o (PARI) a(n) = 2*binomial(9*n+2,n)/(9*n+2);

%o (PARI) {a(n)=local(B=1); for(i=0, n, B=(1+x*B^(9/2))^2+x*O(x^n)); polcoeff(B, n)}

%o (Magma) [2*Binomial(9*n+2, n)/(9*n+2): n in [0..30]];

%Y Cf. A000108, A143554, A234506, A234507, A234508, A234509, A234510, A234513, A232265.

%K nonn

%O 0,2

%A _Tim Fulford_, Dec 27 2013