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A368629 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.

%I A368629 #22 Jan 11 2024 20:54:39
%S A368629 1,1,4,9,88,210,2644,6493,91992,229646,3484008,8789562,139443168,
%T A368629 354379540,5801987316,14824740645,248459660984,637465292438,
%U A368629 10878564788984,28001827694446,484778825103504,1251132971284668,21915195896364296,56682787977509650,1002570518541796720
%N A368629 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 A368629 Conjecture: a(n) is odd when n = 2^k - 1 for k >= 0, and even elsewhere.
%H A368629 Paul D. Hanna, <a href="/A368629/b368629.txt">Table of n, a(n) for n = 0..1200</a>
%F A368629 G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
%F A368629 (1) A(x) = 1 + x*(A(x)^2 + A(-x)^2)/2 + x*(A(x)^4 - A(-x)^4)/2.
%F A368629 (2) A(x) = A(-x) + x*A(x)^2 + x*A(-x)^2.
%F A368629 (3) A(x) = 2 - A(-x) + x*A(x)^4 - x*A(-x)^4.
%F A368629 (4) A(x) = 2 - A(-x) + (A(x) - A(-x))*(A(x)^2 - A(-x)^2).
%F A368629 (5.a) A(x) = (1 - sqrt(1 - 4*x*A(-x) - 4*x^2*A(-x)^2)) / (2*x).
%F A368629 (5.b) A(-x) = (sqrt(1 + 4*x*A(x) - 4*x^2*A(x)^2) - 1) / (2*x).
%F A368629 (6) (A(x) + A(-x))/2 = 1/(1 - (A(x) - A(-x))^2).
%F A368629 (7.a) Sum_{n>=0} a(n) * (sqrt(2) - 1)^n/3^n = sqrt(2).
%F A368629 (7.b) Sum_{n>=0} a(n) * (1 - sqrt(2))^n/3^n = 1.
%e A368629 G.f.: A(x) = 1 + x + 4*x^2 + 9*x^3 + 88*x^4 + 210*x^5 + 2644*x^6 + 6493*x^7 + 91992*x^8 + 229646*x^9 + 3484008*x^10 + 8789562*x^11 + 139443168*x^12 + ...
%e A368629 RELATED SERIES.
%e A368629 A(x)^2 = 1 + 2*x + 9*x^2 + 26*x^3 + 210*x^4 + 668*x^5 + 6493*x^6 + 21538*x^7 + 229646*x^8 + 779772*x^9 + 8789562*x^10 + ...
%e A368629 A(x)^4 = 1 + 4*x + 22*x^2 + 88*x^3 + 605*x^4 + 2644*x^5 + 20114*x^6 + 91992*x^7 + 741154*x^8 + 3484008*x^9 + 29125100*x^10 + ...
%e A368629 The odd bisection of A(x) may be formed from the even bisection of A(x)^2:
%e A368629 (A(x) - A(-x))/2 = x + 9*x^3 + 210*x^5 + 6493*x^7 + 229646*x^9 + ...
%e A368629 (A(x)^2 + A(-x)^2)/2 = 1 + 9*x^2 + 210*x^4 + 6493*x^6 + 229646*x^8 + ...
%e A368629 The even bisection of A(x) may be formed from the odd bisection of A(x)^4:
%e A368629 (A(x) + A(-x))/2 = 1 + 4*x^2 + 88*x^4 + 2644*x^6 + 91992*x^8 + 3484008*x^10 + ...
%e A368629 (A(x)^4 - A(-x)^4)/2 = 4*x + 88*x^3 + 2644*x^5 + 91992*x^7 + 3484008*x^9 + ...
%e A368629 SPECIFIC VALUES.
%e A368629 A(-r) = 1 and A(r) = sqrt(2) at r = (sqrt(2) - 1)/3 = 0.138071....
%o A368629 (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 A368629 A = 1 + x*(A^2 + B^2)/2 + x*(A^4 - B^4)/2 ; ); polcoeff(A, n)}
%o A368629 for(n=0, 30, print1(a(n), ", "))
%Y A368629 Cf. A368633, A368628, A368593, A368626, A368627.
%K A368629 nonn
%O A368629 0,3
%A A368629 _Paul D. Hanna_, Jan 10 2024