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%I #9 Aug 16 2014 22:49:12
%S 1,1,0,2,0,5,0,12,0,20,0,-30,0,-546,0,-3672,0,-18796,0,-79640,0,
%T -270955,0,-584340,0,903396,0,20471948,0,155046180,0,872787888,0,
%U 4012121412,0,14728928136,0,34982326212,0,-40695186320,0,-1194336566976,0,-9612277504606,0,-56604770338290,0
%N G.f. satisfies: A(x) = 1 + x*AGM(A(x)^4, A(-x)^4).
%C Here AGM(x,y) = AGM((x+y)/2, sqrt(x*y)) denotes the arithmetic-geometric mean.
%H Paul D. Hanna, <a href="/A245928/b245928.txt">Table of n, a(n) for n = 0..350</a>
%F G.f. satisfies: A(x) = 1 + x*AGM( (A(x)^4 + A(-x)^4)/2, A(x)^2*A(-x)^2 ).
%e G.f.: A(x) = 1 + x + 2*x^3 + 5*x^5 + 12*x^7 + 20*x^9 - 30*x^11 - 546*x^13 -...
%e where
%e AGM(A(x)^4,A(-x)^4) = 1 + 2*x^2 + 5*x^4 + 12*x^6 + 20*x^8 - 30*x^10 - 546*x^12 -...
%e RELATED SERIES:
%e A(x)^2 = 1 + 2*x + x^2 + 4*x^3 + 4*x^4 + 10*x^5 + 14*x^6 + 24*x^7 + 44*x^8 +...
%e A(x)^4 = 1 + 4*x + 6*x^2 + 12*x^3 + 25*x^4 + 44*x^5 + 92*x^6 + 156*x^7 + 308*x^8 +...
%e (A(x)^4 + A(-x)^4)/2 = 1 + 6*x^2 + 25*x^4 + 92*x^6 + 308*x^8 + 878*x^10 + 1614*x^12 -...
%e A(x)^2*A(-x)^2 = 1 - 2*x^2 - 7*x^4 - 20*x^6 - 44*x^8 - 26*x^10 + 494*x^12 + 4152*x^14 +...
%o (PARI) {a(n)=local(A=1+x);for(i=1,n,A=1 + x*agm(A^4,subst(A,x,-x +x*O(x^n))^4));polcoeff(A,n)}
%o for(n=0,40,print1(a(n),", "))
%K sign
%O 0,4
%A _Paul D. Hanna_, Aug 15 2014
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