A230547 - OEIS

%I #47 Sep 12 2025 10:16:01

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

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

%U 9142452842985,58083251802345,369816259792035,2359448984037600,15082416490309740,96586612269316884,619586741695427928

%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 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. Mathm. 15 (2010), 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: A(x) = {1 + x*A(x)^(p/r)}^r, here p=3, r=9.

%F D-finite with recurrence 2*n*(2*n+9)*(n+4)*a(n) -3*(3*n+7)*(n+2)*(3*n+8)*a(n-1)=0. - _R. J. Mathar_, Nov 22 2024

%F a(n) ~ 3^(3*n+21/2) / (4^(n+5) * n^(3/2) * sqrt(Pi)). - _Amiram Eldar_, Sep 12 2025

%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