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%I #8 Jan 10 2024 23:58:09
%S 1,1,2,14,32,345,810,10492,24880,356252,848992,12946094,30942208,
%T 492621678,1179648292,19379467704,46468665184,781821568212,
%U 1876521420624,32169136799832,77270414837888,1344812759618473,3232175494812466,56957048059132524,136958995341531504
%N Expansion of g.f. A(x) satisfying A(x) = 1 + x*(A(x)^2 - A(-x)^2)/2 + x*(A(x)^4 + A(-x)^4)/2.
%C Conjecture: a(n) is odd when n = (4^k - 1)/3 for k >= 0, and even elsewhere.
%H Paul D. Hanna, <a href="/A368628/b368628.txt">Table of n, a(n) for n = 0..600</a>
%F G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
%F (1) A(x) = 1 + x*(A(x)^2 - A(-x)^2)/2 + x*(A(x)^4 + A(-x)^4)/2.
%F (2) A(x) = 2 - A(-x) + x*A(x)^2 - x*A(-x)^2.
%F (3) A(x) = A(-x) + x*A(x)^4 + x*A(-x)^4.
%F (4.a) A(x) = (1 - sqrt(1-8*x + 4*x*A(-x) + 4*x^2*A(-x)^2)) / (2*x).
%F (4.b) A(-x) = (sqrt(1+8*x - 4*x*A(x) + 4*x^2*A(x)^2) - 1) / (2*x).
%F (5) (A(x) + A(-x))/2 = 1/(1 - x*(A(x) - A(-x))).
%e G.f.: A(x) = 1 + x + 2*x^2 + 14*x^3 + 32*x^4 + 345*x^5 + 810*x^6 + 10492*x^7 + 24880*x^8 + 356252*x^9 + 848992*x^10 + 12946094*x^11 + 30942208*x^12 + ...
%e RELATED SERIES.
%e A(x)^2 = 1 + 2*x + 5*x^2 + 32*x^3 + 96*x^4 + 810*x^5 + 2634*x^6 + 24880*x^7 + 84668*x^8 + 848992*x^9 + 2974649*x^10 + ...
%e A(x)^4 = 1 + 4*x + 14*x^2 + 84*x^3 + 345*x^4 + 2324*x^5 + 10492*x^6 + 74540*x^7 + 356252*x^8 + 2609552*x^9 + 12946094*x^10 + ...
%e The even bisection of A(x) may be formed from the odd bisection of A(x)^2:
%e (A(x) + A(-x))/2 = 1 + 2*x^2 + 32*x^4 + 810*x^6 + 24880*x^8 + 848992*x^10 + ...
%e (A(x)^2 - A(-x)^2)/2 = 2*x + 32*x^3 + 810*x^5 + 24880*x^7 + 848992*x^9 + ...
%e The odd bisection of A(x) may be formed from the even bisection of A(x)^4:
%e (A(x) - A(-x))/2 = x + 14*x^3 + 345*x^5 + 10492*x^7 + 356252*x^9 + ...
%e (A(x)^4 + A(-x)^4)/2 = 1 + 14*x^2 + 345*x^4 + 10492*x^6 + 356252*x^8 + ...
%o (PARI) {a(n) = my(A=1+x,B); for(i=1,n, A=truncate(A)+x*O(x^i); B=subst(A, x, -x);
%o A = 1 + x*(A^2 - B^2)/2 + x*(A^4 + B^4)/2 ; ); polcoeff(A, n)}
%o for(n=0, 30, print1(a(n), ", "))
%Y Cf. A368629, A368593, A368626, A368627.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Jan 10 2024
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