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%I #20 Mar 19 2017 00:15:38
%S 1,1,16,264,4480,77952,1386496,25135616,463233024,8658673664,
%T 163829383168,3132565553152,60446638866432,1175715287400448,
%U 23028562592268288,453848132868898816,8993594212565909504
%N Generalized Catalan numbers.
%C a(n) = K(8,8; n)/8 with K(a,b; n) defined in a comment to A068763.
%H Fung Lam, <a href="/A068770/b068770.txt">Table of n, a(n) for n = 0..750</a>
%F a(n) = (8^n) * p(n, -7/8) with the row polynomials p(n, x) defined from array A068763.
%F a(n+1) = 8*sum(a(k)*a(n-k), k=0..n), n>=1, a(0)=1=a(1).
%F G.f.: (1-sqrt(1-32*x*(1-7*x)))/(16*x).
%F a(n) = (4^(n-1)*14^(1/2*n+1/2)*LegendreP(n+1,2/7*14^(1/2)) - LegendreP(n,2/7*14^(1/2))*4^n*14^(1/2*n))/n for n > 0. - _Mark van Hoeij_, Apr 23 2010
%F Recurrence: (n+1)*a(n) = 224*(2-n)*a(n-2) + 16*(2*n-1)*a(n-1). - _Fung Lam_, Mar 04 2014
%F a(n) ~ sqrt(1+2*sqrt(2)) * (16+4*sqrt(2))^n / (4*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Mar 04 2014
%t CoefficientList[Series[(1-Sqrt[1-32*x*(1-7*x)])/(16*x), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Mar 04 2014 *)
%o (PARI) a(n) = if(n, (4^(n-1)*14^(1/2*n+1/2)*pollegendre(n+1,2/7*14^(1/2)) - pollegendre(n,2/7*14^(1/2))*4^n*14^(n/2))\/n, 1) \\ _Charles R Greathouse IV_, Mar 19 2017
%Y Cf. A000108, A068764-A068769, A068771-A068772, A025227-A025230.
%K nonn,easy
%O 0,3
%A _Wolfdieter Lang_, Mar 04 2002
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