A089813 Expansion of Jacobi theta function (theta_2(q) - 3*theta_2(q^9))/(2 q^(1/4)) in powers of q.

1, 0, -2, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset

0,3

Comments

This is the sequence A089812 with interleaved zeros. - Michael Somos, Nov 21 2017

Links

G. C. Greubel, Table of n, a(n) for n = 0..5000

Eric Weisstein's World of Mathematics, Jacobi Theta Functions

I. J. Zucker, Further Relations Amongst Infinite Series and Products. II. The Evaluation of Three-Dimensional Lattice Sums, J. Phys. A: Math. Gen. 23, 117-132, 1990.

Formula

Euler transform of period 12 sequence [0, -2, 0, -1, 0, -1, 0, -1, 0, -2, 0, -1, ...]. - Michael Somos, Nov 21 2017

Example

G.f. = 1 - 2*x^2 + x^6 + x^12 - 2*x^20 + x^30 + x^42 - 2*x^56 + x^72 + x^90 - 2*x^110 + ...
G.f. = q - 2*q^9 + q^25 + q^49 - 2*q^81 + q^121 + q^169 - 2*q^225 + q^289 + q^361 + ...

Mathematica

A089813[n_] := SeriesCoefficient[(EllipticTheta[2, 0, q] - 3*EllipticTheta[2, 0, q^9])/(2 q^(1/4), {q, 0, n}]; Table[A089813[n], {n, 0, 50}] (* G. C. Greubel, Nov 20 2017 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ q^2]^2 QPochhammer[ q^12] / (QPochhammer[q ^4] QPochhammer[ q^6]), {q, 0, n}]; (* Michael Somos, Nov 21 2017 *)

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^12 + A) / (eta(x^4 + A) * eta(x^6 + A)), n))}; /* Michael Somos, Nov 21 2017 */
(PARI) lista(nn) = {q='q+O('q^nn); Vec(eta(q^2)^2*eta(q^12)/(eta(q^4)*eta(q^6)))} \\ Altug Alkan, Mar 22 2018

Crossrefs

Cf. A089812.

Sequence in context: A368073 A086071 A322212 * A343023 A337760 A037845

Adjacent sequences: A089810 A089811 A089812 * A089814 A089815 A089816

Keyword

sign

Author

Eric W. Weisstein, Nov 12 2003

Status

approved