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%I #25 Sep 08 2022 08:46:06
%S 1,7,63,644,7105,82467,992446,12271512,154962990,1990038435,
%T 25909892008,341225775072,4537563627415,60842326873230,
%U 821692714673340,11167153485624304,152610018401940330,2095863415900961490,28910564819681953485,400379714692751795820
%N a(n) = 7*binomial(6*n + 7, n)/(6*n + 7).
%C Fuss-Catalan sequence is a(n,p,r) = r*binomial(n*p + r, n)/(n*p + r); this is the case p = 6, r = 7.
%H Vincenzo Librandi, <a href="/A233743/b233743.txt">Table of n, a(n) for n = 0..200</a>
%H J-C. Aval, <a href="http://arxiv.org/abs/0711.0906">Multivariate Fuss-Catalan Numbers</a>, arXiv:0711.0906 [math.CO], 2007.
%H J-C. Aval, <a href="http://dx.doi.org/10.1016/j.disc.2007.08.100">Multivariate Fuss-Catalan Numbers</a>, 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 Wikipedia, <a href="https://en.wikipedia.org/wiki/Fuss-Catalan_number">Fuss-Catalan number</a>
%F G.f. satisfies: A(x) = {1 + x*A(x)^(p/r)}^r, where p = 6, r = 7.
%F From _Peter Bala, Oct 16 2015: (Start)
%F O.g.f. A(x) = 1/x * series reversion (x*C(-x)^7), where C(x) = (1 - sqrt(1 - 4*x))/(2*x) is the o.g.f. for the Catalan numbers A000108. See cross-references for other Fuss-Catalan sequences with o.g.f. 1/x * series reversion (x*C(-x)^k), k = 3 through 11.
%F A(x)^(1/7) is the o.g.f. for A002295. (End)
%t Table[7 Binomial[6 n + 7, n]/(6 n + 7), {n, 0, 40}] (* _Vincenzo Librandi_, Dec 16 2013 *)
%o (PARI) a(n) = 7*binomial(6*n+7,n)/(6*n+7);
%o (PARI) {a(n)=local(B=1); for(i=0, n, B=(1+x*B^(6/7))^7+x*O(x^n)); polcoeff(B, n)}
%o (Magma) [7*Binomial(6*n+7, n)/(6*n+7): n in [0..30]]; // _Vincenzo Librandi_, Dec 16 2013
%Y Cf. A000108, A002295, A212071, A212072, A212073, A130564, A233827, A233829, A233830.
%Y Cf. A000245 (k = 3), A006629 (k = 4), A196678 (k = 5), A233668 (k = 6), A233835 (k = 8), A234467 (k = 9), A232265 (k = 10), A229963 (k = 11).
%K nonn,easy
%O 0,2
%A _Tim Fulford_, Dec 15 2013
%E More terms from _Vincenzo Librandi_, Dec 16 2013
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