%I #29 Oct 23 2015 21:00:46

%S 1,1,2,10,44,294,1728,13389,93130,796620,6235288,57551130,493813936,

%T 4857378920,44989814920,468103507718,4633862094852,50749496457992,

%U 533271010341720,6126256486912776,67990630238066888,817168635245112432,9541543704324657008,119719059789052412360

%N G.f. A(x) satisfies: A(x) = Series_Reversion( x - (A(x)^2 + A(-x)^2)/2 ).

%H Paul D. Hanna, <a href="/A227852/b227852.txt">Table of n, a(n) for n = 1..300</a>

%F G.f. A(x) satisfies:

%F (1) A(x) = x + (A(A(x))^2 + A(-A(x))^2)/2.

%F (2) A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) (A(x)^2 + A(-x)^2)^n / (n!*2^n).

%F (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.

%e G.f.: A(x) = x + x^2 + 2*x^3 + 10*x^4 + 44*x^5 + 294*x^6 + 1728*x^7 +...

%e The series reversion of A(x), G(x) where A(G(x)) = x, begins:

%e 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) +...

%e and can be formed from a bisection of A(x)^2:

%e 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 +...

%e 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:

%e F(x) = x^2 - 5*x^4 + 112*x^6 - 4320*x^8 + 227766*x^10 - 14942616*x^12 +...

%o (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)}

%o for(n=1,25,print1(a(n),", "))

%o (PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}

%o {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)}

%o for(n=1,25,print1(a(n),", "))

%Y Cf. A263531, A213591, A141202.

%K nonn

%O 1,3

%A _Paul D. Hanna_, Oct 31 2013