A320992 Expansion of (Product_{k>0} theta_4(q^k)/theta_3(q^k))^(1/2), where theta_3() and theta_4() are the Jacobi theta functions.

1, -2, 0, -2, 6, -2, 4, -6, 8, -16, 8, -14, 26, -26, 24, -30, 58, -50, 60, -78, 90, -118, 104, -138, 192, -224, 204, -268, 366, -354, 412, -474, 596, -694, 724, -818, 1052, -1162, 1176, -1470, 1756, -1918, 2052, -2434, 2814, -3168, 3396, -3806, 4674, -5124, 5396

Offset

0,2

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Eric Weisstein's World of Mathematics, Jacobi Theta Functions

Formula

a(n) = (-1)^n * A320078(n).

Expansion of Product_{k>0} (eta(q^k)^2*eta(q^(4*k))) / eta(q^(2*k))^3.

Expansion of Product_{k>0} theta_4(q^(2*k-1)).

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

Mathematica

nmax = 60; CoefficientList[Series[Product[Sqrt[EllipticTheta[4, 0, x^k] / EllipticTheta[3, 0, x^k]], {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 26 2018 *)

Crossrefs

Convolution inverse of A320968.

Cf. A000122, A002448, A320078, A320970.

Sequence in context: A033727 A033757 A320240 * A320078 A136426 A325199

Adjacent sequences: A320989 A320990 A320991 * A320993 A320994 A320995

Keyword

sign

Author

Seiichi Manyama, Oct 26 2018

Status

approved