%I #49 Sep 08 2022 08:46:06

%S 1,10,75,510,3325,21252,134550,848250,5340060,33622600,211915132,

%T 1337675430,8458829925,53591180360,340185835500,2163581913780,

%U 13786238414025,88004926973250,562763873596575,3604713725613000,23126371951808268,148594788106641360

%N a(n) = 10 * binomial(3*n+10,n)/(3*n+10).

%C Fuss-Catalan sequence is a(n,p,r) = r*binomial(np+r,n)/(np+r), this is the case p=3, r=10.

%H Vincenzo Librandi, <a href="/A233657/b233657.txt">Table of n, a(n) for n = 0..200</a>

%H David Bevan, Robert Brignall, Andrew Elvey Price and Jay Pantone, <a href="http://arxiv.org/abs/1711.10325">A structural characterisation of Av(1324) and new bounds on its growth rate</a>, arXiv preprint arXiv:1711.10325 [math.CO], 2017-2019.

%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="https://web.archive.org/web/20170830003716/http://www.mhhe.com/math/advmath/rosen/r5/instructor/applications/ch07.pdf">Catalan Numbers Chapter 7</a>

%H Wojciech Mlotkowski, <a href="https://www.elibm.org/article/10000158">Fuss-Catalan Numbers in Noncommutative Probability</a>, Docum. Mathm. 15: 939-955 (2010).

%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=10.

%F +2*n*(n+5)*(2*n+9)*a(n) -3*(3*n+7)*(n+3)*(3*n+8)*a(n-1)=0. - _R. J. Mathar_, Feb 16 2018

%F E.g.f.: F([10/3, 11/3, 4], [1, 11/2, 6], 27*x/4), where F is the generalized hypergeometric function. - _Stefano Spezia_, Oct 08 2019

%p A233657:=n->10*binomial(3*n+10,n)/(3*n+10): seq(A233657(n), n=0..20); # _Wesley Ivan Hurt_, Oct 10 2014

%t Table[10 Binomial[3 n + 10, n]/(3 n + 10), {n, 0, 30}]

%o (PARI) a(n) = 10*binomial(3*n+10,n)/(3*n+10);

%o (PARI) {a(n)=local(B=1); for(i=0, n, B=(1+x*B^(3/10))^10+x*O(x^n)); polcoeff(B, n)}

%o (Magma) [10*Binomial(3*n+10, n)/(3*n+10): n in [0..30]];

%Y Column 5 of A092276.

%Y Cf. A000108, A001764, A006013, A006629, A102893, A006630, A102594, A006631, A230547.

%K nonn,easy

%O 0,2

%A _Tim Fulford_, Dec 14 2013