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a(n) = binomial(8*n+2,n)/(4*n+1).
10

%I #43 Sep 08 2022 08:46:06

%S 1,2,17,200,2728,40508,635628,10368072,174047640,2987139122,

%T 52177566870,924548764752,16578073731752,300252605231600,

%U 5484727796499708,100933398334075824,1869468985400220600,34823332479175275600,651947852922093741585

%N a(n) = binomial(8*n+2,n)/(4*n+1).

%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 = 2.

%H Vincenzo Librandi, <a href="/A234461/b234461.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 Gi-Sang Cheon, S.-T. Jin, and L. W. Shapiro, <a href="https://doi.org/10.1016/j.laa.2015.03.015">A combinatorial equivalence relation for formal power series</a>, Linear Algebra and its Applications, Volume 491, 15 February 2016, Pages 123-137.

%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="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 = 2.

%F a(n) = 2*binomial(8n+1,n-1)/n for n>0, a(0)=1. [_Bruno Berselli_, Jan 19 2014]

%F A(x^3) = 1/x * series reversion (x/C(x^3)^2), where C(x) = (1 - sqrt(1 - 4*x))/(2*x) is the o.g.f. for the Catalan numbers A000108. A(x)^(1/2) is the o.g.f. for A007556. - _Peter Bala_, Oct 14 2015

%t Table[Binomial[8 n + 2, n]/(4 n + 1), {n, 0, 30}]

%o (PARI) a(n) = binomial(8*n+2,n)/(4*n+1);

%o (PARI) {a(n)=local(B=1); for(i=0, n, B=(1+x*B^4)^2+x*O(x^n)); polcoeff(B, n)}

%o (Magma) [Binomial(8*n+2, n)/(4*n+1): n in [0..30]];

%Y Cf. A000108, A007556, A234462, A234463, A234464, A234465, A234466, A234467, A230390.

%K nonn,easy

%O 0,2

%A _Tim Fulford_, Dec 26 2013