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A368634 Expansion of g.f. A(x) satisfying A(x) = 1 + x*(3*A(x)^2 - A(-x)^2)/2.

%I A368634 #11 Jan 13 2024 02:31:09
%S A368634 1,1,4,9,52,138,904,2581,18020,53622,389112,1189146,8855560,27571156,
%T A368634 209174544,660249549,5079702852,16203796158,126033559960,405408758062,
%U A368634 3180991167640,10301855821452,81414086371696,265150389430914,2108026107021224,6897985805906972,55119920086104496
%N A368634 Expansion of g.f. A(x) satisfying A(x) = 1 + x*(3*A(x)^2 - A(-x)^2)/2.
%C A368634 Conjecture: a(n) is odd when n = 2^k - 1 for k >= 0 and even elsewhere.
%H A368634 Paul D. Hanna, <a href="/A368634/b368634.txt">Table of n, a(n) for n = 0..600</a>
%F A368634 G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
%F A368634 (1.a) A(x) = 1 + x*(3*A(x)^2 - A(-x)^2)/2.
%F A368634 (1.b) A(x) = 1 + x*(A(x)^2 + A(-x)^2)/2 + 2*x*(A(x)^2 - A(-x)^2)/2.
%F A368634 (2.a) (A(x) + A(-x))/2 = 1 + 2*x*(A(x)^2 - A(-x)^2)/2.
%F A368634 (2.b) (A(x) - A(-x))/2 = x*(A(x)^2 + A(-x)^2)/2.
%F A368634 (2.c) (A(x) + A(-x))/2 = 1/(1 - 2*x*(A(x) - A(-x))).
%F A368634 (3.a) A(x) = (1 - sqrt(1 - 4*x*A(-x) - 4*x^2*A(-x)^2)) / (2*x).
%F A368634 (3.b) A(-x) = (sqrt(1 + 4*x*A(x) - 4*x^2*A(x)^2) - 1) / (2*x).
%F A368634 (4.a) A(x) = (1 - sqrt(1-16*x + 8*x*A(-x) + 16*x^2*A(-x)^2)) / (4*x).
%F A368634 (4.b) A(-x) = (sqrt(1+16*x - 8*x*A(x) + 16*x^2*A(x)^2) - 1) / (4*x).
%F A368634 (5) 0 = (1-2*x) - (1-6*x)*A(x) - x*(3+8*x)*A(x)^2 + 12*x^2*A(x)^3 - 8*x^3*A(x)^4.
%F A368634 (6) x = (1 + 6*x*A(x) - 8*x^2*A(x)^2 - sqrt(1 + 4*x*A(x) - 4*x^2*A(x)^2))/4.
%F A368634 (7) A(x) = (1/x)*Series_Reversion( (1 + 6*x - 8*x^2 - sqrt(1 + 4*x - 4*x^2))/4 ).
%F A368634 (8.a) Sum_{n>=0} a(n) * (sqrt(3) - 1)^n/4^n = sqrt(3).
%F A368634 (8.b) Sum_{n>=0} a(n) * (1 - sqrt(3))^n/4^n = 1.
%e A368634 G.f.: A(x) = 1 + x + 4*x^2 + 9*x^3 + 52*x^4 + 138*x^5 + 904*x^6 + 2581*x^7 + 18020*x^8 + 53622*x^9 + 389112*x^10 + 1189146*x^11 + 8855560*x^12 + ...
%e A368634 RELATED SERIES.
%e A368634 We can see from the expansion of A(x)^2, which begins
%e A368634 A(x)^2 = 1 + 2*x + 9*x^2 + 26*x^3 + 138*x^4 + 452*x^5 + 2581*x^6 + 9010*x^7 + 53622*x^8 + 194556*x^9 + 1189146*x^10 + 4427780*x^11 + 27571156*x^12 + ...
%e A368634 that the odd bisection of A(x) is derived from the even bisection of A(x)^2:
%e A368634 (A(x) - A(-x))/2 = x + 9*x^3 + 138*x^5 + 2581*x^7 + 53622*x^9 + ...
%e A368634 (A(x)^2 + A(-x)^2)/2 = 1 + 9*x^2 + 138*x^4 + 2581*x^6 + 53622*x^8 + ...
%e A368634 and the even bisection of A(x) is derived from the odd bisection of A(x)^2:
%e A368634 (A(x) + A(-x))/2 = 1 + 4*x^2 + 52*x^4 + 904*x^6 + 18020*x^8 + 389112*x^10 + ...
%e A368634 (A(x)^2 - A(-x)^2)/2 = 2*x + 26*x^3 + 452*x^5 + 9010*x^7 + 194556*x^9 + ...
%e A368634 so that (A(x) + A(-x))/2 = 1 + 2*x * (A(x)^2 - A(-x)^2)/2.
%e A368634 SPECIFIC VALUES.
%e A368634 A(-r) = 1 and A(r) = sqrt(3) at r = (sqrt(3) - 1)/4 = 0.183012701892219....
%o A368634 (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 A368634 A = 1 + x*(A^2 + B^2)/2 + 2*x*(A^2 - B^2)/2 ; ); polcoeff(A, n)}
%o A368634 for(n=0, 30, print1(a(n), ", "))
%o A368634 (PARI) {a(n) = my(A); A = (1/x)*serreverse( (1 + 6*x - 8*x^2 - sqrt(1 + 4*x - 4*x^2 +x^2*O(x^n)))/4 ); polcoeff(A, n)}
%o A368634 for(n=0, 30, print1(a(n), ", "))
%Y A368634 Cf. A368633, A368629, A368627.
%K A368634 nonn
%O A368634 0,3
%A A368634 _Paul D. Hanna_, Jan 12 2024