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A272822
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G.f. satisfies: A(x) = 1 + 2*x*AGM(A(x)^2, A(-x)^2).
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1
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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, 6323744, 0, 0, 0, -119299200, 0, 0, 0, 2311546200, 0, 0, 0, -45729024480, 0, 0, 0, 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
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OFFSET
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0,2
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COMMENTS
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Here AGM(x,y) = AGM((x+y)/2, sqrt(x*y)) denotes the arithmetic-geometric mean.
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LINKS
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FORMULA
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G.f. A(x) satisfies:
(1) A(x)*A(-x) = 2 - (A(x)^2 + A(-x)^2)/2.
(2) A(x) = 1 + 2*x*AGM(A(x)*A(-x), (A(x)^2 + A(-x)^2)/2).
(3) A(x) = 1 + 2*x*AGM(A(x)*A(-x), 2 - A(x)*A(-x)).
(4) A(x) = 1 + 2*x*AGM(1, sqrt( A(x)*A(-x) * (A(x)^2 + A(-x)^2)/2 ) ).
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EXAMPLE
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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 +...
where A(x) = 1 + 2*x*AGM(A(x)^2, A(-x)^2).
RELATED SERIES.
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 +...
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 +...
(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 +...
where A(x)*A(-x) = 2 - (A(x)^2 + A(-x)^2)/2.
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PROG
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(PARI) /* From definition: A(x) = 1 + 2*x*AGM(A(x)^2, A(-x)^2) */
{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)}
for(n=0, 60, print1(a(n), ", "))
(PARI) /* From A(x) = 1 + 2*x*AGM(A(x)*A(-x), (A(x)^2 + A(-x)^2)/2) */
{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)}
for(n=0, 60, print1(a(n), ", "))
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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