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%I #14 Jun 29 2019 06:28:18
%S 1,2,4,4,0,-4,0,8,0,-20,0,56,0,-168,0,528,0,-1716,0,5720,0,-19448,0,
%T 67184,0,-235144,0,832048,0,-2971600,0,10697760,0,-38779380,0,
%U 141430680,0,-518579160,0,1910554800,0,-7069052760,0,26256481680,0,-97865068080,0,365930254560,0
%N G.f. satisfies: A(x) = 1 + 2*x*sqrt(A(x)/A(-x)).
%F G.f. A(x) satisfies:
%F (1) A(x) * A(-x) = 1 + 4*x^2.
%F (2) A(x) / A(-x) = 2*A(x) - 1.
%F (3) A(x) + A(-x) = 2 + 8*x^2.
%F (4) A(x) = 1 + 2*x + 4*x^2 + 4*x^3*C(-x^2), where C(x) = 1 + x*C(x)^2 is the g.f. of the Catalan numbers (A000108).
%F a(n) = 4*hypergeom([3-n, 2-n], [2], -1)) for n>0. - _Peter Luschny_, May 22 2017
%F D-recurrence: -(n+2)*(n-1)*a(n) -(n+3)*(n-2)*a(n-1) -4*(n+2)*(n-4)*a(n-2) -4*(n+3)*(n-5)*a(n-3)=0. - _R. J. Mathar_, May 22 2019
%e G.f.: A(x) = 1 + 2*x + 4*x^2 + 4*x^3 - 4*x^5 + 8*x^7 - 20*x^9 + 56*x^11 +...
%e where
%e A(x)/A(-x) = 1 + 4*x + 8*x^2 + 8*x^3 - 8*x^5 + 16*x^7 - 40*x^9 + 112*x^11 +...
%p a := n -> `if`(n=0, 1, 4*hypergeom([3-n, 2-n], [2], -1)):
%p seq(simplify(a(n)), n=0..48); # _Peter Luschny_, May 22 2017
%t m = 50; A[_] = 1; Do[A[x_] = 1 + 2 x Sqrt[A[x]/A[-x]] + O[x]^m // Normal, {m}]; CoefficientList[A[x], x] (* _Jean-François Alcover_, Jun 29 2019 *)
%o (PARI) {a(n)=local(A=1+2*x);for(i=1,n,A=1+2*x*sqrt(A/subst(A,x,-x+x*O(x^n))));polcoeff(A,n)}
%K sign
%O 0,2
%A _Paul D. Hanna_, Oct 29 2011
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