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A068766 Generalized Catalan numbers 4*x*A(x)^2 -A(x)+1-3*x=0.

%I A068766 #20 Feb 03 2025 08:38:28
%S A068766 1,1,8,68,608,5664,54528,538944,5441024,55889408,582348800,6140864512,
%T A068766 65414742016,702897995776,7609805045760,82929151328256,
%U A068766 908978855215104,10014523823357952,110840574196580352,1231847926116384768
%N A068766 Generalized Catalan numbers 4*x*A(x)^2 -A(x)+1-3*x=0.
%C A068766 a(n)=K(4,4; n)/4 with K(a,b; n) defined in a comment to A068763.
%H A068766 Fung Lam, <a href="/A068766/b068766.txt">Table of n, a(n) for n = 0..925</a>
%F A068766 a(n)=(4^n)*p(n, -3/4) with the row polynomials p(n, x) defined from array A068763.
%F A068766 a(n+1)= 4*sum(a(k)*a(n-k), k=0..n), n>=1, a(0)=1=a(1).
%F A068766 G.f.: (1-sqrt(1-16*x*(1-3*x)))/(8*x).
%F A068766 Recurrence: (n+1)*a(n) = 48*(2-n)*a(n-2) + 8*(2*n-1)*a(n-1). - _Fung Lam_, Mar 04 2014
%F A068766 a(n) ~ sqrt(6) * 12^n / (4*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Mar 04 2014
%F A068766 a(n) = 2^n*GegenbauerC(n-1, -n, -2)/(2*n) for n>=1. - _Peter Luschny_, May 09 2016
%p A068766 a := n -> `if`(n=0,1,simplify(2^n*GegenbauerC(n-1, -n, -2))/(2*n)):
%p A068766 seq(a(n), n=0..19); # _Peter Luschny_, May 09 2016
%t A068766 CoefficientList[Series[(1-Sqrt[1-16*x*(1-3*x)])/(8*x), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Mar 04 2014 *)
%Y A068766 Cf. A000108, A068764-5, A068767-72, A025227-30.
%K A068766 nonn,easy
%O A068766 0,3
%A A068766 _Wolfdieter Lang_, Mar 04 2002