A290405 Expansion of (a(q) / b(q))^3 in powers of q where a(), b() are cubic AGM theta functions.

1, 27, 324, 2430, 13716, 64557, 265356, 983556, 3353076, 10670373, 32031288, 91455804, 249948828, 657261999, 1669898592, 4113612864, 9853898292, 23010586596, 52494114852, 117209543940, 256559365656, 551320914321, 1164556135440, 2420715030912, 4956677613180

Offset: 0

Author

Seiichi Manyama, Jul 30 2017

Keywords

• nonn

Comments

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..10000
• J. M. Borwein, P. B. Borwein and F. Garvan, Some Cubic Modular Identities of Ramanujan, Trans. Amer. Math. Soc. 343 (1994), 35-47.

Crossrefs

Cf. A000726, A005882, A005928, A121590, A215690.

Programs

Mathematica

nmax = 20; CoefficientList[Series[1 + 27*x*Product[(1 + x^k + x^(2*k))^12, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 30 2017 *)

Formula

a(n) = 27 * A121590(n) for n > 0.

G.f.: (1 + 9*(eta(q^9)/eta(q))^3)^3 = 1 + 27*(eta(q^3)/eta(q))^12 = 1 + (c(q) / b(q))^3.