A357788 a(n) = coefficient of x^(2*n) in C(x) defined by: C(x) + i*S(x) = Sum_{n=-oo..+oo} i^n * (2*x)^(n^2) * F(x)^n, where F(x) is the g.f. of A357787 such that C(x)^2 + S(x)^2 = 1.

1, 0, -32, -256, -2048, -12288, -32768, 131072, 3276800, 28311552, 125829120, -285212672, -11274289152, -110326972416, -511101108224, 2052994367488, 66383014526976, 707123415613440, 4088396548931584, -1608585511436288, -341992096703447040, -4383726471663845376

Offset

0,3

Links

Paul D. Hanna, Table of n, a(n) for n = 0..300

Formula

Given g.f. C(x) = Sum_{n>=0} a(n)*x^(2*n) and related functions S(x) and F(x) (g.f. of A357787) satisfy the following.

(1) C(x)^2 + S(x)^2 = 1.

(2) C(x) + i*S(x) = Sum_{n=-oo..+oo} i^n * (2*x)^(n^2) * F(x)^n.

(3) 1/F(x) = F(-x).

(4) C(x) = 1 + Sum_{n>=1} (-1)^n * (2*x)^(4*n^2) * (F(x)^(2*n) + F(-x)^(2*n)).

(5) S(x) = Sum_{n>=0} (-1)^n * (2*x)^((2*n+1)^2) * (F(x)^(2*n+1) - F(-x)^(2*n+1)).

(6) C(x) + i*S(x) = Product_{n>=1} (1 + i*(2*x)^(2*n-1)*F(x)) * (1 - i*(2*x)^(2*n-1)/F(x)) * (1 - (2*x)^(2*n)), due to the Jacobi triple product identity.

Example

G.f.: C(x) = 1 - 32*x^4 - 256*x^6 - 2048*x^8 - 12288*x^10 - 32768*x^12 + 131072*x^14 + 3276800*x^16 + 28311552*x^18 + 125829120*x^20 - 285212672*x^22 - 11274289152*x^24 - 110326972416*x^26 - 511101108224*x^28 + 2052994367488*x^30 + 66383014526976*x^32 + ...
such that C(x) and S(x) = sqrt(1 - C(x)^2) form the real and imaginary parts of the doubly infinite sum
C(x) + i*S(x) = Sum_{n=-oo..+oo} i^n * (2*x)^(n^2) * F(x)^n
where F(x) is the g.f. of A357787 and normalizes the given theta series so that C(x)^2 + S(x)^2 = 1.
Explicitly,
C(x) = 1 - (2*x)^4*(F(x)^2 + F(-x)^2) + (2*x)^16*(F(x)^4 + F(-x)^4) - (2*x)^36*(F(x)^6 + F(-x)^6) + (2*x)^64*(F(x)^8 + F(-x)^8) + ... + (-1)^n * (2*x)^(4*n^2) * (F(x)^(2*n) + F(-x)^(2*n)) + ...
S(x) = (2*x)*(F(x) - F(-x))  - (2*x)^9*(F(x)^3 - F(-x)^3) + (2*x)^25*(F(x)^5 - F(-x)^5) - (2*x)^49*(F(x)^7 - F(-x)^7) + ... + (-1)^n * (2*x)^((2*n+1)^2) * (F(x)^(2*n+1) - F(-x)^(2*n+1)) + ...
where
F(x) = 1 + 2*x + 2*x^2 + 8*x^3 + 14*x^4 + 32*x^5 + 68*x^6 + 22*x^8 - 768*x^9 - 2020*x^10 - 9216*x^11 - 23156*x^12 - 45056*x^13 - 115320*x^14 + 32768*x^15 + ... + A357787(n) * x^n + ...
S(x) = 8*x^2 + 32*x^4 + 128*x^6 - 9216*x^10 - 94208*x^12 - 671744*x^14 - 3014656*x^16 + 1245184*x^18 + 171704320*x^20 + 1756364800*x^22 + 8338276352*x^24 + ... + A357789(n)*x^(2*n) + ...
RELATED SERIES.
C(x)^2 = 1 - 64*x^4 - 512*x^6 - 3072*x^8 - 8192*x^10 + 131072*x^12 + 2097152*x^14 + 19136512*x^16 + 115343360*x^18 + 260046848*x^20 - 3791650816*x^22 - 60666413056*x^24 - 471909531648*x^26 - 1563368095744*x^28 + ...
The logarithm of C(x) + i*S(x) is an imaginary even function that begins:
log(C(x) + i*S(x)) = i*(16*x^2/2 + 128*x^4/4 + 1280*x^6/6 + 8192*x^8/8 + 14336*x^10/10 - 81920*x^12/12 - 2129920*x^14/14 - 12582912*x^16/16 + 36044800*x^18/18 + 1722810368*x^20/20 + 20768096256*x^22/22 + ...)
because C(x)^2 + S(x)^2 = 1.
Also observe that the following roots of C(x) + i*S(x) are integer series:
(C(x) + i*S(x))^(1/2) = 1 + 4*i*x^2 + (-8 + 16*i)*x^4 + (-64 + 96*i)*x^6 + (-544 + 384*i)*x^8 + (-3584 - 640*i)*x^10 + (-14592 - 14848*i)*x^12 + (-30720 - 148480*i)*x^14 + (325120 - 757760*i)*x^16 + (3948544 + 104448*i)*x^18 + (17731584 + 47472640*i)*x^20 + (-87916544 + 533676032*i)*x^22 + ...
(C(x) + i*S(x))^(1/4) = 1 + 2*i*x^2 + (-2 + 8*i)*x^4 + (-16 + 52*i)*x^6 + (-138 + 240*i)*x^8 + (-928 + 188*i)*x^10 + (-4052 - 3152*i)*x^12 + (-11744 - 47320*i)*x^14 + (48806 - 244896*i)*x^16 + (770752 + 362860*i)*x^18 + (3352836 + 21880112*i)*x^20 + (-25037664 + 242360920*i)*x^22 + ...

Prog

(PARI) {a(n) = my(F=[1, 2], THETA=1); for(i=1, 2*n, F = concat(F, 0); m=sqrtint(#F+9);
THETA = sum(n=-m, m, I^n * (2*x)^(n^2) * truncate(Ser(F))^n + x*O(x^(#F+2)));
F[#F] = -polcoeff( (real(THETA)^2 + imag(THETA)^2)/64, #F+2)); polcoeff(real(THETA), 2*n)}
for(n=0, 25, print1(a(n), ", "))

Crossrefs

Cf. A357787, A357789, A357806.

Sequence in context: A159982 A195592 A097243 * A022327 A318022 A320407

Adjacent sequences: A357785 A357786 A357787 * A357789 A357790 A357791

Keyword

sign

Author

Paul D. Hanna, Dec 05 2022

Status

approved