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%I #24 Sep 08 2022 08:46:06
%S 1,3,24,253,3045,39627,543004,7718340,112752783,1682460520,
%T 25533901536,392912889915,6116090678334,96133810101609,
%U 1523687678528400,24324750346691480,390786855500604195,6313161418594235271,102494297789621214400,1671366110239940499000
%N a(n) = 3*binomial(7*n+3, n)/(7*n+3).
%C Fuss-Catalan sequence is a(n,p,r) = r*binomial(np+r,n)/(np+r), this is the case p=7, r=3.
%H Vincenzo Librandi, <a href="/A233833/b233833.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 Clemens Heuberger, Sarah J. Selkirk, and Stephan Wagner, <a href="https://arxiv.org/abs/2204.14023">Enumeration of Generalized Dyck Paths Based on the Height of Down-Steps Modulo k</a>, arXiv:2204.14023 [math.CO], 2022.
%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=7, r=3.
%F From _Ilya Gutkovskiy_, Sep 14 2018: (Start)
%F E.g.f.: 6F6(3/7,4/7,5/7,6/7,8/7,9/7; 2/3,5/6,1,7/6,4/3,3/2; 823543*x/46656).
%F a(n) ~ 7^(7*n+5/2)/(sqrt(Pi)*3^(6*n+5/2)*4^(3*n+2)*n^(3/2)). (End)
%t Table[3 Binomial[7 n + 3, n]/(7 n + 3), {n, 0, 30}]
%o (PARI) a(n)=3*binomial(7*n+3,n)/(7*n+3);
%o (PARI) {a(n)=local(B=1); for(i=0, n, B=(1+x*B^(7/3))^3+x*O(x^n)); polcoeff(B, n)}
%o (Magma) [3*Binomial(7*n+3, n)/(7*n+3): n in [0..30]];
%Y Cf. A000108, A002296, A233832, A143547, A233834, A130565, A233835, A233907, A233908.
%K nonn
%O 0,2
%A _Tim Fulford_, Dec 16 2013
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