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%I #23 Jan 24 2024 07:52:10
%S 1,1,6,13,114,290,2892,7901,84090,239222,2648244,7732914,87894324,
%T 261371940,3027588120,9125058525,107215635402,326501869166,
%U 3879094785060,11910103389734,142766337272988,441265565242268,5328172865489448,16559430499708018,201171901999797924
%N Expansion of g.f. A(x) satisfying A(x) = 1 + 2*x*A(x)^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="/A368633/b368633.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 + 2*x*A(x)^2 - x*A(-x)^2.
%F (1.b) A(x) = 1 + x*(A(x)^2 + A(-x)^2)/2 + 3*x*(A(x)^2 - A(-x)^2)/2.
%F (2.a) (A(x) + A(-x))/2 = 1 + 3*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 - 3*x*(A(x) - A(-x))).
%F (3.a) A(x) = (1 - sqrt(1-8*x + 8*x^2*A(-x)^2)) / (4*x).
%F (3.b) A(-x) = (sqrt(1+8*x + 8*x^2*A(x)^2) - 1) / (4*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 = (1-x) - (1-4*x)*A(x) - 2*x*(1+3*x)*A(x)^2 + 12*x^2*A(x)^3 - 9*x^3*A(x)^4.
%F (6) x = (1 + 4*x*A(x) - 6*x^2*A(x)^2 - sqrt(1 + 4*x*A(x) - 4*x^2*A(x)^2))/2.
%F (7) A(x) = (1/x)*Series_Reversion( (1 + 4*x - 6*x^2 - sqrt(1 + 4*x - 4*x^2))/2 ).
%F (8.a) Sum_{n>=0} a(n) * (sqrt(2) - 1)^n/3^n = sqrt(2).
%F (8.b) Sum_{n>=0} a(n) * (1 - sqrt(2))^n/3^n = 1.
%F Conjecture D-finite with recurrence +375*n*(127034*n -380695)*(n-1)*(n+1) *a(n) -50*n*(n-1) *(1761680*n^2 -13025595*n +22423399)*a(n-1) -24*(n-1) *(85874984*n^3 -429099788*n^2 +603573743*n -272338575)*a(n-2) +8*(476358272*n^4 -5427553976*n^3 +22342502584*n^2 -39872302249*n +26255347914)*a(n-3) +864*(2*n-7) *(508136*n^3 -2793120*n^2 +5307886*n -3864543)*a(n-4) -1152*(n-4) *(352336*n-843439) *(2*n-7) *(2*n-9)*a(n-5)=0. - _R. J. Mathar_, Jan 24 2024
%e G.f. A(x) = 1 + x + 6*x^2 + 13*x^3 + 114*x^4 + 290*x^5 + 2892*x^6 + 7901*x^7 + 84090*x^8 + 239222*x^9 + 2648244*x^10 + 7732914*x^11 + 87894324*x^12 + ...
%e RELATED SERIES.
%e We can see from the expansion of A(x)^2, which begins
%e A(x)^2 = 1 + 2*x + 13*x^2 + 38*x^3 + 290*x^4 + 964*x^5 + 7901*x^6 + 28030*x^7 + 239222*x^8 + 882748*x^9 + 7732914*x^10 + 29298108*x^11 + 261371940*x^12 + ...
%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 + 13*x^3 + 290*x^5 + 7901*x^7 + 239222*x^9 + ...
%e (A(x)^2 + A(-x)^2)/2 = 1 + 13*x^2 + 290*x^4 + 7901*x^6 + 239222*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 + 6*x^2 + 114*x^4 + 2892*x^6 + 84090*x^8 + 2648244*x^10 + ...
%e (A(x)^2 - A(-x)^2)/2 = 2*x + 38*x^3 + 964*x^5 + 28030*x^7 + 882748*x^9 + ...
%e so that (A(x) + A(-x))/2 = 1 + 3*x * (A(x)^2 - A(-x)^2)/2.
%e SPECIFIC VALUES.
%e A(-r) = 1 and A(r) = sqrt(2) at r = (sqrt(2) - 1)/3 = 0.138071187457698....
%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 + 3*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 + 4*x - 6*x^2 - sqrt(1 + 4*x - 4*x^2 +x^2*O(x^n)))/2 ); polcoeff(A,n)}
%o for(n=0,30,print1(a(n),", "))
%Y Cf. A368634, A368629, A368627.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Jan 11 2024
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