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A227852
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G.f. A(x) satisfies: A(x) = Series_Reversion( x - (A(x)^2 + A(-x)^2)/2 ).
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3
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1, 1, 2, 10, 44, 294, 1728, 13389, 93130, 796620, 6235288, 57551130, 493813936, 4857378920, 44989814920, 468103507718, 4633862094852, 50749496457992, 533271010341720, 6126256486912776, 67990630238066888, 817168635245112432, 9541543704324657008, 119719059789052412360
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OFFSET
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1,3
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LINKS
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FORMULA
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G.f. A(x) satisfies:
(1) A(x) = x + (A(A(x))^2 + A(-A(x))^2)/2.
(2) A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) (A(x)^2 + A(-x)^2)^n / (n!*2^n).
(3) (A(I*x)^2 + A(-I*x)^2)/2 = -F(x), where F(x) is the g.f. of A263531 and satisfies: F(x) = B(x)^2 - C(x)^2 such that B(x) + I*C(x) = Series_Reversion(x - I*F(x)), where I^2 = -1.
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EXAMPLE
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G.f.: A(x) = x + x^2 + 2*x^3 + 10*x^4 + 44*x^5 + 294*x^6 + 1728*x^7 +...
The series reversion of A(x), G(x) where A(G(x)) = x, begins:
G(x) = x - x^2 - 5*x^4 - 112*x^6 - 4320*x^8 - 227766*x^10 - 14942616*x^12 - 1162657840*x^14 +...+ (-1)^n * A263531(n)*x^(2*n) +...
and can be formed from a bisection of A(x)^2:
A(x)^2 = x^2 + 2*x^3 + 5*x^4 + 24*x^5 + 112*x^6 + 716*x^7 + 4320*x^8 + 32290*x^9 + 227766*x^10 + 1893488*x^11 + 14942616*x^12 + 134816212*x^13 + 1162657840*x^14 +...
The related g.f. of A263531, F(x) = -(A(I*x)^2 + A(-I*x)^2)/2, satisfies: F(x) = B(x)^2 - C(x)^2 such that B(x) + I*C(x) = Series_Reversion(x - I*F(x)), where I^2 = -1:
F(x) = x^2 - 5*x^4 + 112*x^6 - 4320*x^8 + 227766*x^10 - 14942616*x^12 +...
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PROG
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(PARI) {a(n)=local(A=x); for(i=1, n, A=serreverse(x-(A^2+subst(A^2, x, -x +x*O(x^n)))/2)); polcoeff(A, n)}
for(n=1, 25, print1(a(n), ", "))
(PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x+x^2+x*O(x^n)); for(i=1, n, A=x+sum(m=1, n, Dx(m-1, (A^2+subst(A, x, -x)^2)^m/2^m/m!))+x*O(x^n)); polcoeff(A, n)}
for(n=1, 25, 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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