%I
%S 1,3,17,117,887,7131,59661,513897,4526603,40588263,369277481,
%T 3400840845,31644776383,297079014771,2810553697989,26770090211793,
%U 256510096740243,2470981627433919,23916768109136289,232485664356533877
%N Number of sequences of n pairs of balanced brackets of three types, round, square, and angle, such that no square brackets are allowed inside matching round brackets.
%H G. C. Greubel, <a href="/A165976/b165976.txt">Table of n, a(n) for n = 0..500</a>
%F O.g.f.: ( 1  x*G(2*x)  sqrt( (1  x*G(2*x))^2  8*x ) )/(4*x) = (3 + sqrt(18*x)  sqrt(10  136*x + 6*sqrt(18*x)))/(16*x), where G(x)=(1sqrt(14*x))/(2*x) is o.g.f. for Catalan numbers A000108.
%t CoefficientList[Series[(3 + Sqrt[1  8*x]  Sqrt[10  136*x + 6*Sqrt[1  8*x]])/(16*x), {x, 0, 50}], x] (* _G. C. Greubel_, Apr 20 2016 *)
%K nonn
%O 0,2
%A _Max Alekseyev_, Oct 02 2009
