A208385 Expansion of b(q) * c(q) * c(q^2) / 9 in powers of q where b(), c() are cubic AGM theta functions.

1, -2, 0, -2, 6, 0, -4, 4, 0, 6, -24, 0, 8, 8, 0, 4, 18, 0, -16, -12, 0, -24, 24, 0, 7, -16, 0, 8, -6, 0, 44, -8, 0, 18, -24, 0, -34, 32, 0, -12, -66, 0, -40, 48, 0, 24, 120, 0, -33, -14, 0, -16, -54, 0, 72, -16, 0, -6, -48, 0, 50, -88, 0, -8, 48, 0, 8, -36

Offset

1,2

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 = 1..1000

N. Elkies, Complexity of computing expansion of a newform level 18 weight 3 and character [3] - OEIS A116418

Formula

Expansion of eta(q)^2 * eta(q^3)^2 * eta(q^6)^3 / eta(q^2) in powers of q.

Euler transform of period 6 sequence [-2, -1, -4, -1, -2, -6, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (18 t)) = 34992^(1/2) (t/i)^3 g(t) where q = exp(2 Pi i t) and g(t) is g.f. for A122407.

G.f.: x * Product_{k>0} (1 - x^k)^2 * (1 - x^(3*k))^2 * (1 - x^(6*k))^3 / (1 - x^(2*k)).

a(3*n) = 0. a(3*n + 1) = A116418(n). a(3*n + 2) = -2 * A122407(n).

Example

G.f. = q - 2*q^2 - 2*q^4 + 6*q^5 - 4*q^7 + 4*q^8 + 6*q^10 - 24*q^11 + 8*q^13 + ...

Mathematica

eta[q_]:= q^(1/24)*QPochhammer[q]; Rest[CoefficientList[Series[eta[q]^2 *eta[q^3]^2*eta[q^6]^3/eta[q^2], {q, 0, 50}], q]] (* G. C. Greubel, Aug 11 2018 *)

Prog

(PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^3 + A)^2 * eta(x^6 + A)^3 / eta(x^2 + A), n))};

Crossrefs

Cf. A116418, A122407.

Sequence in context: A212085 A265882 A324253 * A186634 A363910 A139213

Adjacent sequences: A208382 A208383 A208384 * A208386 A208387 A208388

Keyword

sign

Author

Michael Somos, Feb 25 2012

Extensions

a(40) corrected by Seiichi Manyama, Jan 09 2017

Status

approved