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%I #36 Sep 08 2022 08:46:06
%S 1,9,108,1488,22230,350244,5729724,96395616,1657248417,28987537150,
%T 514215324216,9229030737264,167283594343320,3057857090083908,
%U 56305821384711720,1043424549990820800,19445145508444588200,364191559218548917713,6851518654436447733980
%N a(n) = 9*binomial(8*n + 9,n)/(8*n + 9).
%C Fuss-Catalan sequence is a(n,p,r) = r*binomial(n*p + r,n)/(n*p + r); this is the case p = 8, r = 9.
%H Vincenzo Librandi, <a href="/A234467/b234467.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="https://www.emis.de/journals/DMJDMV/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 = 8, r = 9.
%F From _Peter Bala_, Oct 16 2015: (Start)
%F O.g.f.: (1/x) * series reversion (x*C(-x)^9), 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)^(1/9) is the o.g.f. for A007556. (End)
%F D-finite with recurrence +7*n*(7*n+3)*(7*n+4)*(7*n+5)*(7*n+6)*(7*n+8)*(7*n+9)*a(n)-128*(2*n+1)*(4*n+1)*(4*n+3)*(8*n+1)*(8*n+3)*(8*n+5)*(8*n+7)*a(n-1) = 0. - _R. J. Mathar_, Feb 09 2020
%F E.g.f.: F([9/8, 5/4, 11/8, 3/2, 13/8, 7/4, 15/8], [1, 10/7, 11/7, 12/7, 13/7, 15/7, 16/7], 16777216*x/823543), where F is the generalized hypergeometric function. - _Stefano Spezia_, Feb 09 2020
%t Table[9 Binomial[8 n + 9, n]/(8 n + 9), {n, 0, 40}] (* _Vincenzo Librandi_, Dec 26 2013 *)
%o (PARI) a(n) = 9*binomial(8*n+9,n)/(8*n+9);
%o (PARI) {a(n)=local(B=1); for(i=0, n, B=(1+x*B^(8/9))^9+x*O(x^n)); polcoeff(B, n)}
%o (Magma) [9*Binomial(8*n+9, n)/(8*n+9): n in [0..30]]; // _Vincenzo Librandi_, Dec 26 2013
%Y Cf. A007556, A234461, A234462, A234463, A234464, A234465, A234466.
%Y Cf. A000108, A000245 (k = 3), A006629 (k = 4), A196678 (k = 5), A233668 (k = 6), A233743 (k = 7), A233835 (k = 8), A232265 (k = 10), A229963 (k = 11).
%K nonn,easy
%O 0,2
%A _Tim Fulford_, Dec 26 2013
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