A320970 Expansion of Product_{k>0} theta_4(q^k)/theta_3(q^k), where theta_3() and theta_4() are the Jacobi theta functions.

1, -4, 4, -4, 20, -28, 20, -52, 84, -104, 156, -180, 308, -460, 468, -684, 1028, -1308, 1592, -2084, 2940, -3668, 4564, -5716, 7556, -9912, 11484, -14616, 19252, -23548, 28316, -35188, 44724, -54532, 65996, -79948, 99784, -122796, 143972, -175372, 216524, -259996, 308004, -371140

Offset

0,2

Links

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Eric Weisstein's World of Mathematics, Jacobi Theta Functions

Formula

Expansion of Product_{k>0} (eta(q^k)^4*eta(q^(4*k))^2) / eta(q^(2*k))^6.

a(n) ~ (-1)^n * exp(Pi*sqrt(log(2)*n)) * (log(2))^(1/4) / (4*n^(3/4)). - Vaclav Kotesovec, Oct 26 2018

Mathematica

With[{nmax=80}, CoefficientList[Series[Product[EllipticTheta[4, 0, q^k]/EllipticTheta[3, 0, q^k], {k, 1, nmax+2}], {q, 0, nmax}], q]] (* G. C. Greubel, Oct 29 2018 *)

Prog

(PARI) m=80; q='q+O('q^m); Vec(1/prod(k=1, m+2, eta(q^(2*k))^6/( eta(q^k)^4* eta(q^(4*k))^2) )) \\ G. C. Greubel, Oct 29 2018

Crossrefs

Cf. A000122, A002448, A320068, A320908, A320967.

Sequence in context: A319257 A131946 A034896 * A216871 A120914 A303397

Adjacent sequences: A320967 A320968 A320969 * A320971 A320972 A320973

Keyword

sign

Author

Seiichi Manyama, Oct 25 2018

Status

approved