A231962 Expansion of b(q)^3 - (1/3)*c(q)^3 in powers of q where b(), c() are cubic AGM theta functions.

1, -18, 0, -90, -234, 0, -216, -900, 0, -738, -1296, 0, -1170, -3060, 0, -1728, -3690, 0, -2160, -6516, 0, -4500, -6480, 0, -3672, -10818, 0, -6570, -11700, 0, -6480, -17316, 0, -8640, -15552, 0, -9594, -24660, 0, -15300, -22032, 0, -10800, -33300, 0, -17280

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 - 9 * q * (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^(1/2) (t/i)^3 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A231961.

a(3*n + 2) = 0. a(3*n + 1) = -18 * A231947(n). a(3*n) = A231961(n).

Example

G.f. = 1 - 18*q - 90*q^3 - 234*q^4 - 216*q^6 - 900*q^7 - 738*q^9 + ...

Mathematica

eta[q_]:= q^(1/24)*QPochhammer[q]; CoefficientList[Series[(eta[q]^3/ eta[q^3])^3 - 9*(eta[q^3]^3/eta[q])^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 / eta(x^3 + A))^3 - 9 * x * (eta(x^3 + A)^3 / eta(x + A))^3, n))}

Crossrefs

Cf. A231947, A231961.

Sequence in context: A289660 A096306 A335187 * A052441 A221394 A025602

Adjacent sequences: A231959 A231960 A231961 * A231963 A231964 A231965

Keyword

sign

Author

Michael Somos, Nov 15 2013

Status

approved