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%I #34 Feb 15 2024 18:45:38
%S 1,9,126,2109,38916,763686,15636192,330237765,7141879503,157366449604,
%T 3520256293710,79735912636302,1825080422272800,42148579533938784,
%U 980892581545169496,22980848343194476245,541581608172776494554,12829884648994115426295,305349921559399354716430
%N a(n) = 9*binomial(10*n+9,n)/(10*n+9).
%C Fuss-Catalan sequence is a(n,p,r) = r*binomial(np+r,n)/(np+r), where p=10, r=9.
%H Vincenzo Librandi, <a href="/A234573/b234573.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; 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 Elżbieta Liszewska and Wojciech Młotkowski, <a href="https://arxiv.org/abs/1907.10725">Some relatives of the Catalan sequence</a>, arXiv:1907.10725 [math.CO], 2019.
%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 J. Sawada, J. Sears, A. Trautrim, and A. Williams, <a href="https://arxiv.org/abs/2308.12405">Demystifying our Grandparent's De Bruijn Sequences with Concatenation Trees</a>, arXiv:2308.12405 [math.CO], 2023.
%F G.f. satisfies: B(x) = {1 + x*B(x)^(p/r)}^r, where p=10, r=9.
%F From _Wolfdieter Lang_, Feb 06 2020: (Start)
%F G.f.: hypergeom([9, 10, ..., 18]/10, [10, 11, ..., 18]/9, (10^10/9^9)*x).
%F E.g.f.: hypergeom([9, 11, 12, ..., 18]/10, [10, 11, ..., 18]/9, (10^10/9^9) * x). Cf. _Ilya Gutkovsky_ in A118971. (End)
%F a(n) = binomial(10*n + 8 , n+1)/(9*n + 8) which is instance k = 9 of c(k, n+1) given in a comment in A130564. x*B(x), with the above given g.f. B(x), is the compositional inverse of y*(1 - y)^9, hence B(x)*(1 - x*B(x))^9 = 1. For another formula for B(x) involving the hypergeometric function 9F8 see the analog formula in A234513. - _Wolfdieter Lang_, Feb 15 2024
%t Table[9 Binomial[10 n + 9, n]/(10 n + 9), {n, 0, 30}]
%o (PARI) a(n) = 9*binomial(10*n+9,n)/(10*n+9);
%o (PARI) {a(n)=local(B=1); for(i=0, n, B=(1+x*B^(10/9))^9+x*O(x^n)); polcoeff(B, n)}
%o (Magma) [9*Binomial(10*n+9, n)/(10*n+9): n in [0..30]];
%Y Cf. A000108, A059968, A118971, A130564, A234513, A234525, A234526, A234527, A234528, A234529, A234570, A234571, A229963.
%K nonn,easy
%O 0,2
%A _Tim Fulford_, Dec 28 2013
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