A068766 Generalized Catalan numbers.

1, 1, 8, 68, 608, 5664, 54528, 538944, 5441024, 55889408, 582348800, 6140864512, 65414742016, 702897995776, 7609805045760, 82929151328256, 908978855215104, 10014523823357952, 110840574196580352, 1231847926116384768

Offset

0,3

Comments

a(n)=K(4,4; n)/4 with K(a,b; n) defined in a comment to A068763.

Links

Fung Lam, Table of n, a(n) for n = 0..925

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

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

Mathematica

CoefficientList[Series[(1-Sqrt[1-16*x*(1-3*x)])/(8*x), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 04 2014 *)

Crossrefs

Cf. A000108, A068764-5, A068767-72, A025227-30.

Sequence in context: A163307 A281337 A152105 * A233736 A279266 A054915

Adjacent sequences: A068763 A068764 A068765 * A068767 A068768 A068769

Keyword

nonn,easy

Author

Wolfdieter Lang, Mar 04 2002

Status

approved