A355867 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.

1, 1, -1, -6, -3, 27, 64, -72, -580, -573, 3276, 10778, -4429, -94493, -153086, 463061, 2197569, 604351, -17222574, -40338277, 64029441, 477897865, 433963667, -3248816635, -10525409672, 6577294016, 106318417880, 163863253517, -599970596239, -2714863450622

Offset

0,4

Comments

What is the pattern to the signs of the terms?

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

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

Links

Paul D. Hanna, Table of n, a(n) for n = 0..180

Formula

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

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

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

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

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

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

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

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

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

Example

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 + ...
Let B = sqrt(A(x)) and i = sqrt(-1), then the imaginary part vanishes in the following sums:
(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 + ...
(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 + ...
(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) + ...
(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) + ...
where
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 + ...
The expansion of Sum_{n=-oo..+oo} x^n * (x^n + i*sqrt(A(x)))^n yields
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 + ...
in which all coefficients of x^n evaluate to zero except the constant term.
Specific values.
Let a = A(1/2) = 1.11275889505675972780876...
and b = sqrt(a) = 1.05487387637421362384214...,
then 2 = Sum_{n=-oo..+oo} 1/2^n * (1/2^n + i*b)^n.
The signs of the terms begin:
[+,+,-,-,-,+,+,-,-,-,+,+,-,-,-,+,+,+,-,-,+,+,+,-,-,+,+,+,-,-,-, +,+,-,-,-,+,+,-,-,-,+,+,-,-,-,+,+,+,-,-,+,+,+,-,-,+,+,+,-,-,-, +,+,-,-,-,+,+,-,-,-,+,+,+,-,-,+,+,+,-,-,+,+,+,-,-,-,+,+,-,-,-, +,+,-,-,-,+,+,+,-,-,+,+,+,-,-,+,+,+,-,-,+,+,+,-,-,-,+,+,-,-,-, +,+,-,-,-,+,+,+,-,-,+,+,+,-,-,+,+,+,-,-,-,+,+,-,-,-,+,+,-,-,-, +,+,+,-,-,+,+,+,-,-,+,+,+,-,-,+,+,+,-,-,-,+,+,-,-,-,+,+,-,-,-, +,+,+,-,-,+,+,+,-,-,+,+,+,-,-, ...].

Prog

(PARI) {a(n) = my(A=[1, 0], B); for(i=1, n, A=concat(A, [0, 0]); B = sqrt(Ser(A));
A[#A-1] = polcoeff( sum(m=-#A, #A, x^m*(x^m + I*B)^m ), #A)/2); A[2*n+1]}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n) = my(A=[1, 0], B); for(i=1, n, A=concat(A, [0, 0]); B = sqrt(Ser(A));
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]}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A355868.

Sequence in context: A294032 A088697 A039631 * A287510 A282418 A287608

Adjacent sequences: A355864 A355865 A355866 * A355868 A355869 A355870

Keyword

sign

Author

Paul D. Hanna, Aug 09 2022

Status

approved