%I #7 Jan 10 2024 23:58:19
%S 1,1,2,7,18,78,220,1043,3090,15402,47044,242126,755076,3973820,
%T 12580344,67303139,215511330,1167556434,3772175860,20640707866,
%U 67167649868,370510806212,1212836703304,6735128062542,22156120392276,123731147310820,408741630687656,2293595176625340
%N Expansion of g.f. A(x) satisfying A(x) = 1 + x*(A(x)^2 - A(-x)^2)/2 + x*sqrt( (A(x)^4 + A(-x)^4)/2 ).
%C Conjecture: a(n) is odd when n = 2^k - 1 for k >= 0, and even elsewhere.
%H Paul D. Hanna, <a href="/A368593/b368593.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*sqrt( (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*sqrt( (A(x)^4 + x*A(-x)^4)/2 ).
%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 + 7*x^3 + 18*x^4 + 78*x^5 + 220*x^6 + 1043*x^7 + 3090*x^8 + 15402*x^9 + 47044*x^10 + 242126*x^11 + 755076*x^12 + ...
%e RELATED SERIES.
%e A(x)^2 = 1 + 2*x + 5*x^2 + 18*x^3 + 54*x^4 + 220*x^5 + 717*x^6 + 3090*x^7 + 10562*x^8 + 47044*x^9 + 165858*x^10 + 755076*x^11 + ...
%e A(x)^4 = 1 + 4*x + 14*x^2 + 56*x^3 + 205*x^4 + 836*x^5 + 3178*x^6 + 13192*x^7 + 51490*x^8 + 216808*x^9 + 862588*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 + 18*x^4 + 220*x^6 + 3090*x^8 + 47044*x^10 + ...
%e (A(x)^2 - A(-x)^2)/2 = 2*x + 18*x^3 + 220*x^5 + 3090*x^7 + 47044*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 + 7*x^3 + 78*x^5 + 1043*x^7 + 15402*x^9 + ...
%e (A(x)^4 + A(-x)^4)/2 = 1 + 14*x^2 + 205*x^4 + 3178*x^6 + 51490*x^8 + ...
%e sqrt( (A(x)^4 + A(-x)^4)/2 ) = 1 + 7*x^2 + 78*x^4 + 1043*x^6 + 15402*x^8 + ...
%o (PARI) {a(n) = my(A=[1]); for(i=1,n,
%o A = Vec(1 + x*(Ser(A)^2 - subst(Ser(A)^2,x,-x))/2 + x*sqrt( (Ser(A)^4 + subst(Ser(A)^4,x,-x))/2 ) +x*O(x^#A) ) );A[n+1]}
%o for(n=0,30,print1(a(n),", "))
%Y Cf. A368628, A368629, A368626, A368627.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Jan 10 2024