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%I #36 May 16 2023 12:10:42
%S 1,2,15,154,1827,23562,320866,4540200,66096459,983592304,14894775896,
%T 228784720710,3555866673450,55819631671902,883738853546472,
%U 14094715154157680,226245021605612955,3652242142988400570,59254515909624764575,965678197027521177200
%N a(n) = 2*binomial(7*n+2,n)/(7*n+2).
%C Fuss-Catalan sequence is a(n,p,r) = r*binomial(np+r,n)/(np+r), this is the case p=7, r=2.
%H Vincenzo Librandi, <a href="/A233832/b233832.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.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Fuss%E2%80%93Catalan_number">Fuss-Catalan number</a>
%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=7, r=2.
%F a(n) = 2*binomial(7n+1, n-1)/n for n>0, a(0)=1. [_Bruno Berselli_, Jan 19 2014]
%F From _Ilya Gutkovskiy_, Sep 14 2018: (Start)
%F E.g.f.: 6F6(2/7,3/7,4/7,5/7,6/7,8/7; 1/2,2/3,5/6,1,7/6,4/3; 823543*x/46656).
%F a(n) ~ 7^(7*n+3/2)/(sqrt(Pi)*3^(6*n+5/2)*4^(3*n+1)*n^(3/2)). (End)
%t Table[2 Binomial[7 n + 2, n]/(7 n + 2), {n, 0, 30}]
%o (PARI) a(n) = 2*binomial(7*n+2,n)/(7*n+2);
%o (PARI) {a(n)=local(B=1); for(i=0, n, B=(1+x*B^(7/2))^2+x*O(x^n)); polcoeff(B, n)}
%o (Magma) [2*Binomial(7*n+2, n)/(7*n+2): n in [0..30]];
%Y Cf. A000108, A002296, A233833 - A233835, A143547, A130565, A233907, A233908.
%K nonn
%O 0,2
%A _Tim Fulford_, Dec 16 2013