A357806 a(n) = coefficient of x^(2*n) in A(x) = 1 + Sum_{n>=1} (-1)^n * x^(4*n^2) * (F(x/2)^(2*n) + F(-x/2)^(2*n)), where F(x) is the g.f. of A357787.

1, 0, -2, -4, -8, -12, -8, 8, 50, 108, 120, -68, -672, -1644, -1904, 1912, 15456, 41160, 59494, -5852, -311040, -996744, -1752680, -840600, 5988928, 24181500, 50438488, 45910304, -103373216, -582387300, -1428882832, -1814475760, 1263429058, 13685575400

Offset

0,3

Comments

a(n) = A357788(n)/4^n for n >= 0.

Links

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

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^(2*n) and related function F(x) (g.f. of A357787) satisfy the following.

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

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

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

Example

G.f.: A(x) = 1 - 2*x^4 - 4*x^6 - 8*x^8 - 12*x^10 - 8*x^12 + 8*x^14 + 50*x^16 + 108*x^18 + 120*x^20 - 68*x^22 - 672*x^24 - 1644*x^26 - 1904*x^28 + 1912*x^30 + 15456*x^32 + 41160*x^34 + 59494*x^36 - 5852*x^38 - 311040*x^40 + ... + A357788(n)*x^(2*n)/4^n + ...
The related function F(x) is the g.f. of A357787 and begins
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 + ...
where A(x) = 1 + Sum_{n>=1} (-1)^n * x^(4*n^2) * (F(x/2)^(2*n) + F(-x/2)^(2*n)).
The square of the g.f. A(x) begins:
A(x)^2 = 1 - 4*x^4 - 8*x^6 - 12*x^8 - 8*x^10 + 32*x^12 + 128*x^14 + 292*x^16 + 440*x^18 + 248*x^20 - 904*x^22 - 3616*x^24 - 7032*x^26 - 5824*x^28 + 13056*x^30 + 66372*x^32 + 146144*x^34 + 145116*x^36 - 250216*x^38 - 1545848*x^40 + ...

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)/4^n}
for(n=0, 35, print1(a(n), ", "))

Crossrefs

Cf. A357787, A357788, A357789.

Sequence in context: A324991 A349228 A337181 * A051165 A377672 A009664

Adjacent sequences: A357803 A357804 A357805 * A357807 A357808 A357809

Keyword

sign

Author

Paul D. Hanna, Dec 05 2022

Status

approved