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A317998
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G.f. A(x) satisfies: Sum_{n>=1} (x + (-1)^n*A(x))^n = 0.
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2
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1, 4, 16, 96, 640, 4480, 32768, 247552, 1915904, 15113216, 121069568, 982269952, 8054767616, 66651537408, 555846729728, 4667099709440, 39420882583552, 334730329587712, 2855644020342784, 24464860907569152, 210393112063770624, 1815582450311495680, 15716738746688733184, 136443973020569042944, 1187652950607087534080, 10362818914536380170240
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OFFSET
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1,2
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LINKS
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FORMULA
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G.f. A(x) satisfies:
(1) A(-A(-x)) = x.
(2a) 0 = Sum_{n>=1} (x + (-1)^n*A(x))^n.
(2b) 1 = 1/(1 - (x + A(x))^2) + (x - A(x))/(1 - (x - A(x))^2).
(3) 0 = x*(1-x)*(1+x)^2 - (1-x)^2*A(x) + (1 + x + 2*x^2)*A(x)^2 + A(x)^3 - A(x)^4.
(4) A(x) = F(F(x)) where Sum_{n>=1} (-1)^n * (F(x) - (-1)^n*F(-x))^n = 0, and F(x) is the g.f. of A317999.
(5a) A(x) = x + 4 * Series_Reversion( x/sqrt(1 + 4*x^2 - 16*x^4) - 2*x^2 )^2.
(5b) A( x/sqrt(1 + 4*x^2 - 16*x^4) - 2*x^2 ) = x/sqrt(1 + 4*x^2 - 16*x^4) + 2*x^2.
a(n) ~ sqrt((-3*r^2 - 4*r^3 + (1+s)^2 + 2*r*(1 - s + 2*s^2)) / (1 + r + 2*r^2 + 3*s - 6*s^2)) / (2*sqrt(Pi) * n^(3/2) * r^(n - 1/2)), where r = 0.10802903243795080465651783320471389432467958205527... and s = 0.28488427575880229259717798521096697574952476989148... are real roots of the system of equations s^2*(1 + r + 2*r^2 + s) + (1-r)*r*(1+r)^2 = (1-r)^2 * s + s^4, s*(2 + 2*r + 4*r^2 + 3*s) = (1-r)^2 + 4*s^3. - Vaclav Kotesovec, Aug 26 2018
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EXAMPLE
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O.g.f.: A(x) = x + 4*x^2 + 16*x^3 + 96*x^4 + 640*x^5 + 4480*x^6 + 32768*x^7 + 247552*x^8 + 1915904*x^9 + 15113216*x^10 + ...
such that
0 = (x - A(x)) + (x + A(x))^2 + (x - A(x))^3 + (x + A(x))^4 + (x - A(x))^5 + (x + A(x))^6 + (x - A(x))^7 + (x + A(x))^8 + ...
RELATED SERIES.
G.f. A(x) = F(F(x)) where
F(x) = x + 2*x^2 + 4*x^3 + 24*x^4 + 112*x^5 + 608*x^6 + 3392*x^7 + 19456*x^8 + 114688*x^9 + 681984*x^10 + 4120576*x^11 + 25182208*x^12 + ... + A317999(n)*x^n + ...
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PROG
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(PARI) {a(n) = my(A=[1]); for(i=1, n, A = concat(A, 0); A[#A] = polcoeff(sum(m=1, #A, (x + (-1)^m*x*Ser(A))^m), #A)); A[n]}
for(n=1, 30, print1(a(n), ", "))
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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