A231948 Expansion of a(q)^2 * b(q) in powers of q where a(), b() are cubic AGM theta functions.

1, 9, 0, -90, 117, 0, -216, 450, 0, -738, 648, 0, -1170, 1530, 0, -1728, 1845, 0, -2160, 3258, 0, -4500, 3240, 0, -3672, 5409, 0, -6570, 5850, 0, -6480, 8658, 0, -8640, 7776, 0, -9594, 12330, 0, -15300, 11016, 0, -10800, 16650, 0, -17280, 14256, 0, -18450

Offset

0,2

Comments

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

Links

G. C. Greubel, Table of n, a(n) for n = 0..2500

Formula

Expansion of (eta(q)^3 + 9 * eta(q^9)^3)^2 * (eta(q) / eta(q^3))^3 in powers of q.

G.f. is a period 1 Fourier series which satisfies f(-1 / (9 t)) = 3^(11/2) (t/i)^3 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A231947.

a(3*n + 2) = 0. a(3*n + 1) = 9 * A231947(n). 3 * A109041(n) = a(3*n) + A109041(3*n) + A181976(3*n).

Example

G.f. = 1 + 9*q - 90*q^3 + 117*q^4 - 216*q^6 + 450*q^7 - 738*q^9 + ...

Mathematica

eta[q_]:= q^(1/24)*QPochhammer[q]; CoefficientList[Series[(eta[q]^3 + 9*eta[q^9]^3)^2*(eta[q]/eta[q^3])^3, {q, 0, 50}], q] (* G. C. Greubel, Aug 08 2018 *)

Prog

(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A)^3 + 9 * x * eta(x^9 + A)^3)^2 * (eta(x + A) / eta(x^3 + A))^3, n))}

Crossrefs

Cf. A109041, A181976, A231947.

Sequence in context: A343575 A270010 A167319 * A222396 A222516 A057403

Adjacent sequences: A231945 A231946 A231947 * A231949 A231950 A231951

Keyword

sign

Author

Michael Somos, Nov 15 2013

Status

approved