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A233666 a(n) = 2*binomial(4*n + 8, n)/(n + 2).

%I A233666 #28 Nov 29 2024 23:50:28
%S A233666 1,8,60,456,3542,28080,226548,1855040,15380937,128896456,1090119316,
%T A233666 9292881360,79769043900,688915123680,5981962494852,52193342019456,
%U A233666 457367224685012,4023551800087200,35521420783728880,314608026125871720,2794654131668318430
%N A233666 a(n) = 2*binomial(4*n + 8, n)/(n + 2).
%C A233666 Fuss-Catalan sequence is a(n,p,r) = r*binomial(np+r,n)/(np+r), this is the case p=4, r=8.
%H A233666 Vincenzo Librandi, <a href="/A233666/b233666.txt">Table of n, a(n) for n = 0..200</a>
%H A233666 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 A233666 Thomas A. Dowling, <a href="http://www.mhhe.com/math/advmath/rosen/r5/instructor/applications/ch07.pdf">Catalan Numbers Chapter 7</a>
%H A233666 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 A233666 G.f. satisfies: B(x) = {1 + x*B(x)^(p/r)}^r, where p=4, r=8.
%F A233666 From _Ilya Gutkovskiy_, Sep 14 2018: (Start)
%F A233666 E.g.f.: 4F4(2,9/4,5/2,11/4; 1,3,10/3,11/3; 256*x/27).
%F A233666 a(n) ~ 2^(8*n+35/2)/(sqrt(Pi)*3^(3*n+17/2)*n^(3/2)). (End)
%F A233666 D-finite with recurrence 3*(3*n+7)*(n+2)*(3*n+8)*a(n) -2*(n+1)*(317*n^2+954*n+709)*a(n-1) +112*(4*n+1)*(2*n+1)*(4*n+3)*a(n-2)=0. - _R. J. Mathar_, Nov 22 2024
%t A233666 Table[2/(n + 2) Binomial[4 n + 8, n], {n, 0, 40}] (* _Vincenzo Librandi_, Dec 14 2013 *)
%o A233666 (PARI) a(n) = 4*binomial(4*n+8,n)/(n+2);
%o A233666 (PARI) {a(n)=local(B=1); for(i=0, n, B=(1+x*B^(1/2))^8+x*O(x^n)); polcoeff(B, n)}
%o A233666 (Magma) [2*Binomial(4*n+8,n)/(n+2): n in [0..30]]; // _Vincenzo Librandi_, Dec 14 2013
%Y A233666 Cf. A000108, A002293, A069271, A006632, A196678, A006633, A233658, A006634, A233667, A006635.
%K A233666 nonn,easy
%O A233666 0,2
%A A233666 _Tim Fulford_, Dec 14 2013