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A068766 Generalized Catalan numbers. 4

%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

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Last modified March 29 06:57 EDT 2024. Contains 371265 sequences. (Running on oeis4.)