A320239 Expansion of theta_3(q) * theta_3(q^3) * theta_3(q^5), where theta_3() is the Jacobi theta function.

1, 2, 0, 2, 6, 2, 4, 4, 4, 14, 0, 0, 14, 4, 4, 0, 6, 12, 8, 4, 2, 20, 0, 4, 20, 2, 8, 10, 12, 4, 4, 4, 16, 32, 0, 0, 26, 4, 0, 12, 0, 20, 8, 4, 8, 6, 4, 4, 42, 18, 0, 8, 20, 12, 16, 0, 12, 48, 8, 8, 0, 16, 8, 12, 14, 0, 16, 4, 20, 24, 4, 0, 36, 28, 0, 2, 20, 8, 8, 4, 6

Offset

0,2

Comments

Also the number of integer solutions (a_1, a_2, a_3) to the equation a_1^2 + 3*a_2^2 + 5*a_3^2 = n.

Links

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Simon Plouffe, Numbers in the base e^Pi, arXiv:2509.15609 [math.NT], 2025. See p. 21/24, marked 214.

Eric Weisstein's World of Mathematics, Jacobi Theta Functions

Formula

Empirical: Sum_{n>=0} a(n) / exp(n*Pi) = -(1/6400) * 3^(1/2) * 5^(3/4) * Pi^(1/4) * Gamma(2/3) * Gamma(7/12)^3 * Gamma(11/12)^2 * (1+3^(1/2))^3 * (-2+3^(1/2)) * (5-5^(1/2))^(3/2) * (5^(1/2)+1)^3 / Gamma(3/4)^8 = A389055. - Simon Plouffe, Sep 22 2025

Crossrefs

Product_{k=1..m} theta_3(q^(2*k-1)): A000122 (m=1), A033716 (m=2), this sequence (m=3), A320240 (m=4).

Cf. A320078, A389055.

Sequence in context: A388481 A373167 A242860 * A033727 A033757 A320240

Adjacent sequences: A320236 A320237 A320238 * A320240 A320241 A320242

Keyword

nonn

Author

Seiichi Manyama, Oct 08 2018

Status

approved