A089805 Expansion of Jacobi theta function (theta_4(q^6) - theta_4(q^(2/3)))/2/q^(2/3).

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

Offset

0,1

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

G. E. Andrews, An introduction to Ramanujan's "lost" notebook, Amer. Math. Monthly 86 (1979), no. 2, 89-108.

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

a(2*n) = A089801(n). a(2*n + 1) = 0. - Michael Somos, Jun 30 2015

From Peter Bala, Feb 15 2021: (Start)

G.f.: Sum_{n >= 0} (-1)^n*q^(6*n^2+4*n)*(1 - q^(4*n+2)) = 1 - x^2 - x^10 + x^16 + x^32 - x^42 - x^66 + + - - ....

Note the identity of Ramanujan: Sum_{n >= 0} q^n/Product_{k = 0..n} 1 + q^(2*k+1) = Sum_{n >= 0} (-1)^n*q^(6*n^2+4*n)*(1 + q^(4*n+2)) = 1 + x^2 - x^10 - x^16 + x^32 + x^42 - - + + .... See Andrews, equation 1.2. (End)

Mathematica

A089805[n_] := SeriesCoefficient[(EllipticTheta[4, 0, q^6] - EllipticTheta[4, 0, q^(2/3)])/(2*q^(2/3)), {q, 0, n}]; Table[A089805[n], {n, 0, 50}] (* G. C. Greubel, Nov 20 2017 *)

Crossrefs

Cf. A089801.

Sequence in context: A016391 A016378 A323513 * A080116 A016359 A014029

Adjacent sequences: A089802 A089803 A089804 * A089806 A089807 A089808

Keyword

sign,easy

Author

Eric W. Weisstein, Nov 12 2003

Status

approved