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%I
%S 1,2,14,132,1430,16796,208012,2674440,35357670,477638700,6564120420,
%T 91482563640,1289904147324,18367353072152,263747951750360,
%U 3814986502092304,55534064877048198,812944042149730764,11959798385860453492,176733862787006701400
%N Catalan numbers with even index (A000108(2*n), n >= 0): a(n) = C(4*n,2*n)/(2*n+1).
%C With interpolated zeros, this is C(n)(1+(-1)^n)/2 with g.f. given by 2/(sqrt(1+4x)+sqrt(1-4x)). - _Paul Barry_, Sep 09 2004
%D G. Markowsky, A method for deriving hypergeometric and related identities from the H^2 Hardy norm of conformal maps, Arxiv preprint arXiv:1205.2458, 2012. - _N. J. A. Sloane_, Oct 18 2012
%H Vincenzo Librandi, <a href="/A048990/b048990.txt">Table of n, a(n) for n = 0..200</a>
%F G.f.: A(x) = sqrt(1/8*x^-1*(1-sqrt(1-16*x))).
%F G.f.: 2F1( (1/4, 3/4); (3/2))(16*x). - _Olivier Gérard_ Feb 17 2011
%F n*(2*n+1)*a(n) -2*(4*n-1)*(4*n-3)*a(n-1)=0. - _R. J. Mathar_, Nov 30 2012
%F E.g.f: 2F2(1/4, 3/4; 1, 3/2; 16*x). - _Vladimir Reshetnikov_, Apr 24 2013
%e sqrt(2*x^-1*(1-sqrt(1-x))) = 1 + 1/8*x + 7/128*x^2 + 33/1024*x^3 + ...
%t f[n_] := CatalanNumber[ 2n]; Array[f, 18, 0] (* Or *)
%t CoefficientList[ Series[ Sqrt[2]/Sqrt[1 + Sqrt[1 - 16 x]], {x, 0, 17}], x] (* _Robert G. Wilson v_ *)
%o (Mupad) combinat::dyckWords::count(2*n) $ n = 0..28 - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 14 2007
%Y Cf. A000108, A024492.
%Y Equals 2 * A065097.
%Y Cf. A000108.
%K easy,nonn
%O 0,2
%A _Wolfdieter Lang_
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