This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A234513 #32 Feb 17 2024 07:01:44
%S A234513 1,8,100,1496,24682,433160,7932196,149846840,2898753715,57135036024,
%T A234513 1143315429776,23166186450680,474347963242860,9799792252101016,
%U A234513 204022381037886400,4276098770070159096,90151561242584838605,1910564646571462338800
%N A234513 8*binomial(9*n+8,n)/(9*n+8).
%C A234513 Fuss-Catalan sequence is a(n,p,r) = r*binomial(np+r,n)/(np+r), where p=9, r=8.
%H A234513 Vincenzo Librandi, <a href="/A234513/b234513.txt">Table of n, a(n) for n = 0..200</a>
%H A234513 J-C. Aval, <a href="http://arxiv.org/pdf/0711.0906v1.pdf">Multivariate Fuss-Catalan Numbers</a>, arXiv:0711.0906v1, Discrete Math., 308 (2008), 4660-4669.
%H A234513 Thomas A. Dowling, <a href="http://www.mhhe.com/math/advmath/rosen/r5/instructor/applications/ch07.pdf">Catalan Numbers Chapter 7</a>.
%H A234513 Elżbieta Liszewska, Wojciech Młotkowski, <a href="https://arxiv.org/abs/1907.10725">Some relatives of the Catalan sequence</a>, arXiv:1907.10725 [math.CO], 2019.
%H A234513 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.
%F A234513 G.f. satisfies: B(x) = {1 + x*B(x)^(p/r)}^r, where p=9, r=8.
%F A234513 From _Wolfdieter Lang_, Feb 06 2020: (Start)
%F A234513 G.f.: hypergeom([8, 9, ..., 16]/9, [9, 10, ..., 16]/8, (9^9/8^8)*x).
%F A234513 E,g,f.: hypergeom([8, 10, 11, ..., 16]/9, [9, 10,..., 16]/8, (9^9/8^8)*x). Cf. _Ilya Gutkovsky_ in A118971. (End)
%F A234513 D-finite with recurrence 128*(8*n+3)*(4*n+3)*(8*n+1)*(2*n+1)*(8*n+7)*(4*n+1)*(8*n+5)*(n+1)*a(n) -81*(9*n+2)*(9*n+4)*(3*n+2)*(9*n-1)*(9*n+1)*(3*n+1)*(9*n+5)*(9*n+7)*a(n-1)=0. - _R. J. Mathar_, Aug 01 2022
%F A234513 From _Wolfdieter Lang_, Feb 15 2024: (Start)
%F A234513 a(n) = binomial(9*n+7, n+1)/(8*n+7), which is instance k = 8 of c(k, n+1) given in A130564.
%F A234513 The g.f. given above, and called B in the first line above, satisfies B(x)*(1 - x*B(x))^8 = 1. For the analog proof of the equivalence see A234466. x*B(x) is the compositional inverse of y*(1 - y)^8.
%F A234513 Another formula for the g.f. is B(x) = (8/(9*x))*(1 - 8F7([-1,1,2,3,4,5,6.7]/9, [1,2,3,4,5,6.7]/8; (9^9/8^8)*x)). (End)
%t A234513 Table[8 Binomial[9 n + 8, n]/(9 n + 8), {n, 0, 30}] (* _Vincenzo Librandi_, Dec 28 2013 *)
%o A234513 (PARI) a(n) = 8*binomial(9*n+8,n)/(9*n+8);
%o A234513 (PARI) {a(n)=local(B=1); for(i=0, n, B=(1+x*B^(9/8))^8+x*O(x^n)); polcoeff(B, n)}
%o A234513 (Magma) [8*Binomial(9*n+8, n)/(9*n+8): n in [0..30]]; // _Vincenzo Librandi_, Dec 28 2013
%Y A234513 Cf. A000108, A118971, A130564, A143554, A234466, A234505, A234506, A234507, A234508, A234509, A234510, A232265.
%K A234513 nonn,easy
%O A234513 0,2
%A A234513 _Tim Fulford_, Dec 27 2013