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A258053
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.
3
1, 1, 1, 0, 1, -2, -4, 0, -7, 20, 42, 0, 84, -272, -584, 0, -1239, 4270, 9288, 0, 20370, -72860, -159840, 0, -358092, 1312824, 2897400, 0, 6587944, -24580512, -54490848, 0, -125256887, 473534676, 1053346410, 0, 2441648384, -9325376200, -20800103016, 0, -48534848222, 186886877456, 417769301220, 0
OFFSET
0,6
FORMULA
a(4*n+3) = 0 for n>=0.
G.f. A(x) satisfies:
(1) (A(x) + x/A(x))^2 = x / Series_Reversion( x/AGM(1,1-8*x) ),
(2) A(x) = sqrt( x / Series_Reversion(x*G(x)) ),
(3) A(x)^2 = G(x/A(x)^2),
(4) A(x*G(x))^2 = G(x),
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.
EXAMPLE
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 +...
RELATED SERIES.
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 +...
satisfies
A(x)^2 = G(x/A(x)^2) and A(x*G(x))^2 = G(x), where
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 +...
Also,
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 +...
and
(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 +...
thus
A(x) = sqrt( (F-2*x + sqrt(F)*sqrt(F-4*x))/2 ) where F is the g.f. of A158100.
PROG
(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)}
for(n=0, 50, print1(a(n), ", "))
CROSSREFS
KEYWORD
sign
AUTHOR
Paul D. Hanna, May 17 2015
STATUS
approved