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%I #29 Sep 08 2022 08:46:06
%S 1,10,135,2100,35475,632502,11714745,223198440,4346520750,86128357150,
%T 1731030945644,35202562937100,723029038312230,14976976398326250,
%U 312522428615310000,6563314391270476752,138617681440915119975,2942332729799060033100,62735156704285184848950
%N a(n) = 10*binomial(9*n + 10, n)/(9*n + 10).
%C Fuss-Catalan sequence is a(n,p,r) = r*binomial(n*p + r,n)/(n*p + r), where p = 9, r = 10.
%H Vincenzo Librandi, <a href="/A232265/b232265.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 = 9, r = 10.
%F From _Peter Bala, Oct 16 2015: (Start)
%F O.g.f. A(x) = 1/x * series reversion (x*C(-x)^10), 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)^(1/10) is the o.g.f. for A062994. (End)
%F D-finite with recurrence: 128*n*(8*n+3)*(4*n+3)*(8*n+9)*(2*n+1)*(8*n+7)*(4*n+5)*(8*n+5)*a(n) -81*(9*n+2)*(9*n+4)*(3*n+2)*(9*n+8)*(9*n+1)*(3*n+1)*(9*n+5)*(9*n+7)*a(n-1)=0. - _R. J. Mathar_, Feb 21 2020
%t Table[10 Binomial[9 n + 10, n]/(9 n + 10), {n, 0, 30}]
%o (PARI) a(n) = 10*binomial(9*n+10,n)/(9*n+10);
%o (PARI) {a(n)=local(B=1); for(i=0, n, B=(1+x*B^(9/10))^10+x*O(x^n)); polcoeff(B, n)}
%o (Magma) [10*Binomial(9*n+10, n)/(9*n+10): n in [0..30]];
%Y Cf. A000108, A143554, A234505, A234506, A234507, A234508, A234509, A234510, A234513.
%Y Cf. A062994, A000245 (k = 3), A006629 (k = 4), A196678 (k = 5), A233668 (k = 6), A233743 (k = 7), A233835 (k = 8), A234467 (k = 9), A229963 (k = 11).
%K nonn,easy
%O 0,2
%A _Tim Fulford_, Dec 28 2013
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