%I #14 Aug 29 2019 12:40:23

%S 1,1,-2,7,-26,102,-420,1787,-7794,34666,-156636,716982,-3317700,

%T 15494156,-72935624,345701843,-1648489762,7902956738,-38067806892,

%U 184152092450,-894259126540,4357738501844,-21302682030328,104439435098718,-513390992000340,2529846489669412,-12494572784556440,61838364112438732,-306647601790749384,1523380558254732312,-7580755340625743760,37783723921640161923

%N O.g.f. A(x) satisfies: 0 = Sum_{n>=1} (b(n) - A(x))^n * (2*x)^n / n, where b(n) = 1 if n is odd or b(n) = 2 if n is even.

%C a(n) is odd iff n = 2^k - 1 for k >= 0.

%C Signed version of A307413.

%H Paul D. Hanna, <a href="/A326564/b326564.txt">Table of n, a(n) for n = 0..520</a>

%F O.g.f. A = A(x) satisfies:

%F (1) 0 = Sum_{n>=1} (3 + (-1)^n - 2*A(x))^n * x^n / n.

%F (2) 0 = arctanh(2*x - 2*x*A) - log(1 - 4*x^2*(2 - A)^2)/2.

%F (3) 1 - 4*x^2*(2 - A)^2 = (1 + 2*x - 2*x*A) / (1 - 2*x + 2*x*A).

%F (4) A(x) = 1 + (A - 2)^2*x + 2*(A - 1)*(A - 2)^2*x^2.

%F (5) 0 = 2*(A - 1)*(A - 2)^2*x^2 + (A - 2)^2*x - (A - 1).

%F (6) x = ( sqrt( (A-2)^4 + 8*(A-1)^2*(A-2)^2 ) - (A-2)^2 ) / (4*(A-1)*(A-2)^2).

%F (7) A(x) = 2 - (1/x) * Series_Reversion( x + x^2/(1 - 2*x^2) ).

%e O.g.f.: A(x) = 1 + x - 2*x^2 + 7*x^3 - 26*x^4 + 102*x^5 - 420*x^6 + 1787*x^7 - 7794*x^8 + 34666*x^9 - 156636*x^10 + 716982*x^11 - 3317700*x^12 + 15494156*x^13 - 72935624*x^14 + 345701843*x^15 - 1648489762*x^16 + ...

%e such that

%e 0 = (1 - A(x))*(2*x) + (2 - A(x))^2*(2*x)^2/2 + (1 - A(x))^3*(2*x)^3/3 + (2 - A(x))^4*(2*x)^4/4 + (1 - A(x))^5*(2*x)^5/5 + (2 - A(x))^6*(2*x)^6/6 + (1 - A(x))^7*(2*x)^7/7 + (2 - A(x))^8*(2*x)^8/8 + (1 - A(x))^9*(2*x)^9/9 + ...

%e SPECIAL ARGUMENTS.

%e A( (3 - sqrt(17))/6 ) = 1/2.

%e A( (15 - sqrt(513))/40 ) = 1/3.

%e ODD TERMS.

%e The odd numbers occur at positions 2^n-1 and begin

%e [1, 1, 7, 1787, 345701843, 37783723921640161923, 1297226675901009799785880946943488094880739, 4359630365907394639251834255689265800511483817161978056491648421720696612963282942355107, ...].

%o (PARI) /* By definition */

%o {a(n) = my(A=[1]); for(i=0, n, A = concat(A, 0); A[#A] = polcoeff(sum(m=1, #A, ( ((m+1)%2) + 1 - Ser(A) )^m * (2*x)^m/m), #A)/2); A[n+1]}

%o for(n=0, 32, print1(a(n), ", "))

%o (PARI) /* From: A(x) = 2 - (1/x) * Series_Reversion( x + x^2/(1 - 2*x^2) ) */

%o {a(n) = my(A = 2 - (1/x)*serreverse(x + x^2/(1 - 2*x^2 +x*O(x^n)))); polcoeff(A,n)}

%o for(n=0, 32, print1(a(n), ", "))

%Y Cf. A316363, A307413.

%K sign

%O 0,3

%A _Paul D. Hanna_, Aug 28 2019