%I #33 Sep 08 2022 08:46:06

%S 1,9,63,408,2565,15939,98670,610740,3786588,23535820,146710476,

%T 917263152,5752004349,36174046743,228124619100,1442387942520,

%U 9142452842985,58083251802345,369816259792035,2359448984037600

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

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

%H Vincenzo Librandi, <a href="/A230547/b230547.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="http://www.math.uiuc.edu/documenta/vol-15/28.pdf">Fuss-Catalan Numbers in Noncommutative Probability</a>, Docum. Mathm. 15: 939-955.

%H Emanuele Munarini, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL20/Munarini/muna4.html">Shifting Property for Riordan, Sheffer and Connection Constants Matrices</a>, Journal of Integer Sequences, Vol. 20 (2017), Article 17.8.2.

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

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

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

%o (PARI) {a(n)=local(B=1); for(i=0, n, B=(1+x*B^(3/9))^9+x*O(x^n)); polcoeff(B, n)}

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

%Y Cf. A000108, A001764, A006013, A006629, A102893, A006630, A102594, A006631, A233657.

%K nonn

%O 0,2

%A _Tim Fulford_, Oct 23 2013