A068766 Generalized Catalan numbers 4*x*A(x)^2 -A(x)+1-3*x=0.
1, 1, 8, 68, 608, 5664, 54528, 538944, 5441024, 55889408, 582348800, 6140864512, 65414742016, 702897995776, 7609805045760, 82929151328256, 908978855215104, 10014523823357952, 110840574196580352, 1231847926116384768
Offset: 0
Links
- Fung Lam, Table of n, a(n) for n = 0..925
Programs
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Maple
a := n -> `if`(n=0,1,simplify(2^n*GegenbauerC(n-1, -n, -2))/(2*n)): seq(a(n), n=0..19); # Peter Luschny, May 09 2016
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Mathematica
CoefficientList[Series[(1-Sqrt[1-16*x*(1-3*x)])/(8*x), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 04 2014 *)
Formula
a(n)=(4^n)*p(n, -3/4) with the row polynomials p(n, x) defined from array A068763.
a(n+1)= 4*sum(a(k)*a(n-k), k=0..n), n>=1, a(0)=1=a(1).
G.f.: (1-sqrt(1-16*x*(1-3*x)))/(8*x).
Recurrence: (n+1)*a(n) = 48*(2-n)*a(n-2) + 8*(2*n-1)*a(n-1). - Fung Lam, Mar 04 2014
a(n) ~ sqrt(6) * 12^n / (4*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Mar 04 2014
a(n) = 2^n*GegenbauerC(n-1, -n, -2)/(2*n) for n>=1. - Peter Luschny, May 09 2016
Comments