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A272822 G.f. satisfies: A(x) = 1 + 2*x*AGM(A(x)^2, A(-x)^2).

%I A272822 #5 May 07 2016 09:00:06
%S A272822 1,2,0,0,0,-8,0,0,0,88,0,0,0,-1248,0,0,0,20056,0,0,0,-347360,0,0,0,
%T A272822 6323744,0,0,0,-119299200,0,0,0,2311546200,0,0,0,-45729024480,0,0,0,
%U A272822 919817083680,0,0,0,-18755086062720,0,0,0,386776930629408,0,0,0,-8053207409464448,0,0,0,169063213304845696,0,0,0,-3574582279957007872,0,0,0,76052003974889011032,0,0,0,-1626996714019973193696,0,0,0,34977427889190951942176,0,0,0,-755249894672423411380352,0,0,0
%N A272822 G.f. satisfies: A(x) = 1 + 2*x*AGM(A(x)^2, A(-x)^2).
%C A272822 Here AGM(x,y) = AGM((x+y)/2, sqrt(x*y)) denotes the arithmetic-geometric mean.
%F A272822 G.f. A(x) satisfies:
%F A272822 (1) A(x)*A(-x) = 2 - (A(x)^2 + A(-x)^2)/2.
%F A272822 (2) A(x) = 1 + 2*x*AGM(A(x)*A(-x), (A(x)^2 + A(-x)^2)/2).
%F A272822 (3) A(x) = 1 + 2*x*AGM(A(x)*A(-x), 2 - A(x)*A(-x)).
%F A272822 (4) A(x) = 1 + 2*x*AGM(1, sqrt( A(x)*A(-x) * (A(x)^2 + A(-x)^2)/2 ) ).
%e A272822 G.f.: A(x) = 1 + 2*x - 8*x^5 + 88*x^9 - 1248*x^13 + 20056*x^17 - 347360*x^21 + 6323744*x^25 - 119299200*x^29 + 2311546200*x^33 - 45729024480*x^37 +...
%e A272822 where A(x) = 1 + 2*x*AGM(A(x)^2, A(-x)^2).
%e A272822 RELATED SERIES.
%e A272822 A(x)^2 = 1 + 4*x + 4*x^2 - 16*x^5 - 32*x^6 + 176*x^9 + 416*x^10 - 2496*x^13 - 6400*x^14 + 40112*x^17 + 107936*x^18 - 694720*x^21 - 1929984*x^22 +...
%e A272822 A(x)*A(-x) = 1 - 4*x^2 + 32*x^6 - 416*x^10 + 6400*x^14 - 107936*x^18 + 1929984*x^22 - 35940096*x^26 + 689571840*x^30 - 13537204640*x^34 +...
%e A272822 (A(x)^2 + A(-x)^2)/2 = 1 + 4*x^2 - 32*x^6 + 416*x^10 - 6400*x^14 + 107936*x^18 - 1929984*x^22 + 35940096*x^26 - 689571840*x^30 + 13537204640*x^34 +...
%e A272822 where A(x)*A(-x) = 2 - (A(x)^2 + A(-x)^2)/2.
%o A272822 (PARI) /* From definition: A(x) = 1 + 2*x*AGM(A(x)^2, A(-x)^2) */
%o A272822 {a(n)=local(A=1+4*x + x*O(x^n)); for(i=1, n, A = 1 + 2*x*agm(A^2, subst(A,x,-x)^2)); polcoeff(A, n)}
%o A272822 for(n=0,60,print1(a(n),", "))
%o A272822 (PARI) /* From A(x) = 1 + 2*x*AGM(A(x)*A(-x), (A(x)^2 + A(-x)^2)/2) */
%o A272822 {a(n)=local(A=1+4*x + x*O(x^n)); for(i=1, n, A = 1 + 2*x*agm(A*subst(A,x,-x), (A^2 + subst(A,x,-x)^2)/2)); polcoeff(A, n)}
%o A272822 for(n=0,60,print1(a(n),", "))
%Y A272822 Cf. A245928, A272823.
%K A272822 sign
%O A272822 0,2
%A A272822 _Paul D. Hanna_, May 07 2016