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Coefficients in the even function A(x) = Sum_{n>=0} a(n)*x^(2*n) such that: 2 = Sum_{n=-oo..+oo} x^n * (x^n + i*sqrt(A(x)))^n, where i^2 = -1.
2

%I #9 Aug 11 2022 19:26:40

%S 1,1,-1,-6,-3,27,64,-72,-580,-573,3276,10778,-4429,-94493,-153086,

%T 463061,2197569,604351,-17222574,-40338277,64029441,477897865,

%U 433963667,-3248816635,-10525409672,6577294016,106318417880,163863253517,-599970596239,-2714863450622

%N Coefficients in the even function A(x) = Sum_{n>=0} a(n)*x^(2*n) such that: 2 = Sum_{n=-oo..+oo} x^n * (x^n + i*sqrt(A(x)))^n, where i^2 = -1.

%C What is the pattern to the signs of the terms?

%C Related identity: Sum_{n=-oo..+oo} (-x)^n * (x^n + y)^n = 0 for all y.

%C Related identity: Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 + y*x^n)^n = 0 for all y.

%H Paul D. Hanna, <a href="/A355867/b355867.txt">Table of n, a(n) for n = 0..180</a>

%F G.f. A(x) = Sum_{n>=0} a(n)*x^(2*n) satisfies the following sums.

%F (1) 2 = Sum_{n=-oo..+oo} x^n * (x^n + i*sqrt(A(x)))^n.

%F (2) 0 = Sum_{n=-oo..+oo} (-1)^n * x^n * (x^n + i*sqrt(A(x)))^n.

%F (3) 1 = Sum_{n=-oo..+oo} x^(2*n) * (x^(2*n) + i*sqrt(A(x)))^(2*n).

%F (4) 1 = Sum_{n=-oo..+oo} x^(2*n+1) * (x^(2*n+1) + i*sqrt(A(x)))^(2*n+1).

%F (5) 2 = Sum_{n=-oo..+oo} x^(n*(n-1)) / (1 + i*sqrt(A(x))*x^n)^n.

%F (6) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 + i*sqrt(A(x))*x^n)^n.

%F (7) 1 = Sum_{n=-oo..+oo} x^(2*n*(2*n-1)) / (1 + i*sqrt(A(x))*x^(2*n))^(2*n).

%F (8) 1 = Sum_{n=-oo..+oo} x^(2*n*(2*n+1)) / (1 + i*sqrt(A(x))*x^(2*n+1))^(2*n+1).

%e G.f.: A(x) = 1 + x^2 - x^4 - 6*x^6 - 3*x^8 + 27*x^10 + 64*x^12 - 72*x^14 - 580*x^16 - 573*x^18 + 3276*x^20 + 10778*x^22 - 4429*x^24 + ...

%e Let B = sqrt(A(x)) and i = sqrt(-1), then the imaginary part vanishes in the following sums:

%e (1) 2 = ... + x^(-3)/(x^(-3) + i*B)^3 + x^(-2)/(x^(-2) + i*B)^2 + x^(-1)/(x^(-1) + i*B) + 1 + x*(x + i*B) + x^2*(x^2 + i*B)^2 + x^3*(x^3 + i*B)^3 + ... + x^n*(x^n + i*sqrt(A(x)))^n + ...

%e (2) 0 = ... - x^(-3)/(x^(-3) + i*B)^3 + x^(-2)/(x^(-2) + i*B)^2 - x^(-1)/(x^(-1) + i*B) + 1 - x*(x + i*B) + x^2*(x^2 + i*B)^2 - x^3*(x^3 + i*B)^3 + ... + (-x)^n*(x^n + i*sqrt(A(x)))^n + ...

%e (3) 1 = ... + x^(-6)/(x^(-6) + i*B)^6 + x^(-4)/(x^(-4) + i*B)^4 + x^(-2)/(x^(-2) + i*B)^2 + 1 + x^2*(x^2 + i*B)^2 + x^4*(x^4 + i*B)^4 + x^6*(x^6 + i*B)^6 + ... + x^(2*n)*(x^(2*n) + i*sqrt(A(x)))^(2*n) + ...

%e (4) 1 = ... + x^(-5)/(x^(-5) + i*B)^5 + x^(-3)/(x^(-3) + i*B)^3 + x^(-1)/(x^(-1) + i*B) + x*(x + i*B) + x^3*(x^3 + i*B)^3 + x^5*(x^5 + i*B)^5 + ... + x^(2*n+1)*(x^(2*n+1) + i*sqrt(A(x)))^(2*n+1) + ...

%e where

%e B = sqrt(A(x)) = 1 + 2*(x/2)^2 - 10*(x/2)^4 - 172*(x/2)^6 - 90*(x/2)^8 + 12284*(x/2)^10 + 90812*(x/2)^12 - 664088*(x/2)^14 - 14660346*(x/2)^16 - 35699220*(x/2)^18 + 1460864084*(x/2)^20 + ...

%e The expansion of Sum_{n=-oo..+oo} x^n * (x^n + i*sqrt(A(x)))^n yields

%e 2 = 2 + (2*i^2 + 2)*x^2 + (2*i^4 + 2*i^2)*x^4 + (2*i^6 + 4*i^4 + 4*i^2 + 2)*x^6 + (2*i^8 + 6*i^6 - 2*i^4 - 6*i^2)*x^8 + (2*i^10 + 8*i^8 - 18*i^4 - 12*i^2)*x^10 + (2*i^12 + 10*i^10 + 4*i^8 - 46*i^6 - 14*i^4 + 30*i^2 + 2)*x^12 + (2*i^14 + 12*i^12 + 10*i^10 - 64*i^8 - 76*i^6 + 110*i^4 + 122*i^2)*x^14 + (2*i^16 + 14*i^14 + 18*i^12 - 80*i^10 - 178*i^8 + 210*i^6 + 308*i^4 + 6*i^2)*x^16 + ...

%e in which all coefficients of x^n evaluate to zero except the constant term.

%e Specific values.

%e Let a = A(1/2) = 1.11275889505675972780876...

%e and b = sqrt(a) = 1.05487387637421362384214...,

%e then 2 = Sum_{n=-oo..+oo} 1/2^n * (1/2^n + i*b)^n.

%e The signs of the terms begin:

%e [+,+,-,-,-,+,+,-,-,-,+,+,-,-,-,+,+,+,-,-,+,+,+,-,-,+,+,+,-,-,-, +,+,-,-,-,+,+,-,-,-,+,+,-,-,-,+,+,+,-,-,+,+,+,-,-,+,+,+,-,-,-, +,+,-,-,-,+,+,-,-,-,+,+,+,-,-,+,+,+,-,-,+,+,+,-,-,-,+,+,-,-,-, +,+,-,-,-,+,+,+,-,-,+,+,+,-,-,+,+,+,-,-,+,+,+,-,-,-,+,+,-,-,-, +,+,-,-,-,+,+,+,-,-,+,+,+,-,-,+,+,+,-,-,-,+,+,-,-,-,+,+,-,-,-, +,+,+,-,-,+,+,+,-,-,+,+,+,-,-,+,+,+,-,-,-,+,+,-,-,-,+,+,-,-,-, +,+,+,-,-,+,+,+,-,-,+,+,+,-,-, ...].

%o (PARI) {a(n) = my(A=[1,0],B); for(i=1,n, A=concat(A,[0,0]); B = sqrt(Ser(A));

%o A[#A-1] = polcoeff( sum(m=-#A,#A, x^m*(x^m + I*B)^m ), #A)/2);A[2*n+1]}

%o for(n=0,30,print1(a(n),", "))

%o (PARI) {a(n) = my(A=[1,0],B); for(i=1,n, A=concat(A,[0,0]); B = sqrt(Ser(A));

%o A[#A-1] = polcoeff( sum(m=-#A,#A, x^(2*m*(2*m-1)) / (1 + I*B*x^(2*m))^(2*m) ), #A));A[2*n+1]}

%o for(n=0,30,print1(a(n),", "))

%Y Cf. A355868.

%K sign

%O 0,4

%A _Paul D. Hanna_, Aug 09 2022