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%I #13 Jan 25 2024 08:45:36
%S 1,1,10,21,310,762,12820,33805,607550,1667214,31182540,87799362,
%T 1686609820,4835044372,94676506920,275037241149,5463738069390,
%U 16035014605830,322140216214300,953095126595062,19320606147948820,57539265876939756,1175037853461723160,3518503980453113106
%N Expansion of g.f. A(x) satisfying A(x) = 1 + 3*x*A(x)^2 - 2*x*A(-x)^2.
%C Conjecture: a(n) is odd when n = 2^k - 1 for k >= 0, and even elsewhere.
%H Paul D. Hanna, <a href="/A368635/b368635.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) A(x) = 1 + 3*x*A(x)^2 - 2*x*A(-x)^2.
%F (1.b) A(x) = 1 + x*(A(x)^2 + A(-x)^2)/2 + 5*x*(A(x)^2 - A(-x)^2)/2.
%F (2.a) (A(x) + A(-x))/2 = 1 + 5*x*(A(x)^2 - A(-x)^2)/2.
%F (2.b) (A(x) - A(-x))/2 = x*(A(x)^2 + A(-x)^2)/2.
%F (2.c) (A(x) + A(-x))/2 = 1/(1 - 5*x*(A(x) - A(-x))).
%F (3.a) A(x) = (1 - sqrt(1 - 40*x + 20*x*A(-x) + 100*x^2*A(-x)^2))/(10*x).
%F (3.b) A(-x) = (sqrt(1 + 40*x - 20*x*A(x) + 100*x^2*A(x)^2) - 1)/(10*x).
%F (4.a) A(x) = (1 - sqrt(1 - 4*x*A(-x) - 4*x^2*A(-x)^2))/(2*x).
%F (4.b) A(-x) = (sqrt(1 + 4*x*A(x) - 4*x^2*A(x)^2) - 1) / (2*x).
%F (5) 0 = (2 - x) - 2*(1-3*x)*A(x) - x*(3+10*x)*A(x)^2 + 30*x^2*A(x)^3 - 25*x^3*A(x)^4.
%F (6) x = (1 + 3*x*A(x) - 5*x^2*A(x)^2) - sqrt(1 + 4*x*A(x) - 4*x^2*A(x)^2).
%F (7) A(x) = (1/x)*Series_Reversion( (1 + 3*x - 5*x^2) - sqrt(1 + 4*x - 4*x^2) ).
%F (8.a) Sum_{n>=0} a(n) * (sqrt(6) - 2)^n/5^n = sqrt(6)/2.
%F (8.b) Sum_{n>=0} a(n) * (2 - sqrt(6))^n/5^n = 1.
%F D-finite with recurrence +10935*n*(n-1)*(75132*n-217883) *(n+1)*a(n) -1458*n*(n-1) *(865728*n^2 -6402143*n+11031849)*a(n-1) -360 *(n-1)*(159580368*n^3 -781944228*n^2 +1079436906*n -451430219)*a(n-2) +24*(3677612544*n^4 -41906753640*n^3 +172755991440*n^2 -308392913875*n +202512185406)*a(n-3) +12000*(2*n-7) *(300528*n^3 -1622852*n^2 +3317670*n -2937917)*a(n-4) -3200*(n-4)*(865728*n -2073503) *(2*n-7)*(2*n-9) *a(n-5)=0. - _R. J. Mathar_, Jan 25 2024
%e G.f.: A(x) = 1 + x + 10*x^2 + 21*x^3 + 310*x^4 + 762*x^5 + 12820*x^6 + 33805*x^7 + 607550*x^8 + 1667214*x^9 + 31182540*x^10 + ...
%e RELATED SERIES.
%e We can see from the expansion of A(x)^2, which begins
%e A(x)^2 = 1 + 2*x + 21*x^2 + 62*x^3 + 762*x^4 + 2564*x^5 + 33805*x^6 + 121510*x^7 + 1667214*x^8 + 6236508*x^9 + 87799362*x^10 + ...
%e that the odd bisection of A(x) is derived from the even bisection of A(x)^2:
%e (A(x) - A(-x))/2 = x + 21*x^3 + 762*x^5 + 33805*x^7 + 1667214*x^9 + ...
%e (A(x)^2 + A(-x)^2)/2 = 1 + 21*x^2 + 762*x^4 + 33805*x^6 + 1667214*x^8 + ...
%e and the even bisection of A(x) is derived from the odd bisection of A(x)^2:
%e (A(x) + A(-x))/2 = 1 + 10*x^2 + 310*x^4 + 12820*x^6 + 607550*x^8 + 31182540*x^10 + ...
%e (A(x)^2 - A(-x)^2)/2 = 2*x + 62*x^3 + 2564*x^5 + 121510*x^7 + 6236508*x^9 + ...
%e so that (A(x) + A(-x))/2 = 1 + 5*x * (A(x)^2 - A(-x)^2)/2.
%e SPECIFIC VALUES.
%e A(-r) = 1 and A(r) = sqrt(6)/2 at r = (sqrt(6) - 2)/5 = 0.0898979485566356....
%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 + 5*x*(A^2 - B^2)/2 ); polcoeff(A, n)}
%o for(n=0,30, print1(a(n),", "))
%o (PARI) {a(n) = my(A); A = (1/x)*serreverse( (1 + 3*x - 5*x^2) - sqrt(1 + 4*x - 4*x^2 +x^2*O(x^n)) ); polcoeff(A, n)}
%o for(n=0, 30, print1(a(n), ", "))
%Y Cf. A368633, A368634, A368629, A368627.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Jan 12 2024
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