A231961 Expansion of b(q)^3 - 3*c(q)^3 in powers of q where b(), c() are cubic AGM theta functions.

1, -90, -216, -738, -1170, -1728, -2160, -4500, -3672, -6570, -6480, -8640, -9594, -15300, -10800, -17280, -18450, -20736, -19656, -32580, -22464, -36900, -32400, -38016, -36720, -54090, -36720, -59058, -58500, -60480, -53136, -86580, -58968, -86400, -77760

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 / eta(q^3))^3 - 81 * (eta(q^3)^3 / eta(q))^3 in powers of q.

G.f. is a period 1 Fourier series which satisfies f(-1 / (3 t)) = - 3^(5/2) (t/i)^3 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A231962.

a(n) = A231948(3*n) = A231962(3*n).

Example

G.f. = 1 - 90*q - 216*q^2 - 738*q^3 - 1170*q^4 - 1728*q^5 - 2160*q^6 + ...

Mathematica

eta[q_]:= q^(1/24)*QPochhammer[q]; CoefficientList[Series[(eta[q]^3/ eta[q^3])^3 - 81*(eta[q^3]^3/eta[q])^3, {q, 0, 50}], q] (* G. C. Greubel, Aug 08 2018 *)

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A)^3 / eta(x^3 + A))^3 - 81 * x * (eta(x^3 + A)^3 / eta(x + A))^3, n))};

Crossrefs

Cf. A231948, A231962.

Sequence in context: A044803 A235081 A074213 * A237131 A363729 A255784

Adjacent sequences: A231958 A231959 A231960 * A231962 A231963 A231964

Keyword

sign

Author

Michael Somos, Nov 15 2013

Status

approved