%I #18 May 09 2016 13:11:47
%S 1,1,8,68,608,5664,54528,538944,5441024,55889408,582348800,6140864512,
%T 65414742016,702897995776,7609805045760,82929151328256,
%U 908978855215104,10014523823357952,110840574196580352,1231847926116384768
%N Generalized Catalan numbers.
%C a(n)=K(4,4; n)/4 with K(a,b; n) defined in a comment to A068763.
%H Fung Lam, <a href="/A068766/b068766.txt">Table of n, a(n) for n = 0..925</a>
%F a(n)=(4^n)*p(n, -3/4) with the row polynomials p(n, x) defined from array A068763.
%F a(n+1)= 4*sum(a(k)*a(n-k), k=0..n), n>=1, a(0)=1=a(1).
%F G.f.: (1-sqrt(1-16*x*(1-3*x)))/(8*x).
%F Recurrence: (n+1)*a(n) = 48*(2-n)*a(n-2) + 8*(2*n-1)*a(n-1). - _Fung Lam_, Mar 04 2014
%F a(n) ~ sqrt(6) * 12^n / (4*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Mar 04 2014
%F a(n) = 2^n*GegenbauerC(n-1, -n, -2)/(2*n) for n>=1. - _Peter Luschny_, May 09 2016
%p a := n -> `if`(n=0,1,simplify(2^n*GegenbauerC(n-1, -n, -2))/(2*n)):
%p seq(a(n), n=0..19); # _Peter Luschny_, May 09 2016
%t CoefficientList[Series[(1-Sqrt[1-16*x*(1-3*x)])/(8*x), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Mar 04 2014 *)
%Y Cf. A000108, A068764-5, A068767-72, A025227-30.
%K nonn,easy
%O 0,3
%A _Wolfdieter Lang_, Mar 04 2002