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

0,2

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.

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.

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 *)

Crossrefs

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

Sequence in context: A224375 A010832 A022719 * A048709 A268973 A160223

Adjacent sequences: A290402 A290403 A290404 * A290406 A290407 A290408

Keyword

nonn

Author

Seiichi Manyama, Jul 30 2017

Status

approved