%I #15 May 22 2015 17:58:28
%S 1,1,1,0,1,-2,-4,0,-7,20,42,0,84,-272,-584,0,-1239,4270,9288,0,20370,
%T -72860,-159840,0,-358092,1312824,2897400,0,6587944,-24580512,
%U -54490848,0,-125256887,473534676,1053346410,0,2441648384,-9325376200,-20800103016,0,-48534848222,186886877456,417769301220,0
%N G.f. A(x) satisfies: 1 = Sum_{n>=0} x^n / (A(x)^2 - x)^(2*n+1) * [ Sum_{k=0..n} C(n,k)^2 * x^k * A(x)^(n-2*k) ]^2.
%F a(4*n+3) = 0 for n>=0.
%F G.f. A(x) satisfies:
%F (1) (A(x) + x/A(x))^2 = x / Series_Reversion( x/AGM(1,1-8*x) ),
%F (2) A(x) = sqrt( x / Series_Reversion(x*G(x)) ),
%F (3) A(x)^2 = G(x/A(x)^2),
%F (4) A(x*G(x))^2 = G(x),
%F where G(x) = 1 / AGM((1+x)^2, 1-6*x+x^2) is the g.f. of A227845, and AGM(x,y) = AGM((x+y)/2,sqrt(x*y)) denotes the arithmetic-geometric mean.
%e G.f.: A(x) = 1 + x + x^2 + x^4 - 2*x^5 - 4*x^6 - 7*x^8 + 20*x^9 + 42*x^10 + 84*x^12 - 272*x^13 - 584*x^14 - 1239*x^16 +...
%e RELATED SERIES.
%e A(x)^2 = 1 + 2*x + 3*x^2 + 2*x^3 + 3*x^4 - 2*x^5 - 10*x^6 - 12*x^7 - 21*x^8 + 22*x^9 + 106*x^10 +...
%e satisfies
%e A(x)^2 = G(x/A(x)^2) and A(x*G(x))^2 = G(x), where
%e G(x) = 1 + 2*x + 7*x^2 + 28*x^3 + 125*x^4 + 590*x^5 + 2891*x^6 + 14536*x^7 + 74497*x^8 +...+ A227845(n)*x^n +...
%e Also,
%e A(x) + x/A(x) = 1 + 2*x + 2*x^4 - 4*x^5 - 16*x^8 + 40*x^9 + 200*x^12 - 544*x^13 - 3006*x^16 +...+ A158122(n)*x^n +...
%e and
%e (A(x) + x/A(x))^2 = 1 + 4*x + 4*x^2 + 4*x^4 - 16*x^6 - 28*x^8 + 176*x^10 + 336*x^12 - 2496*x^14 +...+ A158100(n)*x^n +...
%e thus
%e A(x) = sqrt( (F-2*x + sqrt(F)*sqrt(F-4*x))/2 ) where F is the g.f. of A158100.
%o (PARI) {a(n)=local(A); A=sqrt(x/serreverse(x/agm((1+x)^2, 1-6*x+x^2 +x^2*O(x^n)))); polcoeff(A, n)}
%o for(n=0, 50, print1(a(n), ", "))
%Y Cf. A227845, A158122, A158100.
%K sign
%O 0,6
%A _Paul D. Hanna_, May 17 2015