A089802 Expansion of q^(-1/3) * (theta_4(q^3) - theta_4(q^(1/3))) / 2 in powers of q.

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

Offset

0,1

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Number 10 of the 14 primitive eta-products which are holomorphic modular forms of weight 1/2 listed by D. Zagier on page 30 of "The 1-2-3 of Modular Forms". - Michael Somos, May 04 2016

Links

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

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

Expansion of q^(-1/3) * (eta(q) * eta(q^6)^2) / (eta(q^2) * eta(q^3)) in powers of q. - Michael Somos, Apr 12 2005

Expansion of chi(-x) * psi(x^3) in powers of x where psi(), chi() are Ramanujan theta functions. - Michael Somos, Dec 23 2011

Expansion of f(-x, -x^5) in powers of x, where f(, ) is Ramanujan's general theta function.

a(n) = b(3*n + 1) where b() is multiplicative with b(3^e) = 0^e, b(2^e) = - (1 + (-1)^e) / 2 if e>0, b(p^e) = (1 + (-1)^e) / 2 if p>3.

G.f. is a period 1 Fourier series which satisfies f(-1 / (144 t)) = 8^(1/2) (t/i)^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A089812. - Michael Somos, Dec 23 2011

Euler transform of period 6 sequence [-1, 0, 0, 0, -1, -1, ...]. - Michael Somos, Apr 12 2005

abs(a(n)) is the characteristic function of A001082. - Michael Somos, Oct 31 2005

G.f.: Sum_{k in Z} (-1)^k * x^((3*k^2 - 2*k)) = Product_{k>0} (1 - x^(6*k)) * (1 - x^(6*k - 1)) * (1 - x^(6*k - 5)). - Michael Somos, Oct 31 2005

A002448(3*n + 1) = -2 * a(n). - Michael Somos, Jul 07 2006

a(n) = (-1)^n * A089801(n).

a(n) = -(1/n)*Sum_{k=1..n} A284362(k)*a(n-k), a(0) = 1. - Seiichi Manyama, Mar 25 2017

Example

G.f. = 1 - x - x^5 + x^8 + x^16 - x^21 - x^33 + x^40 + x^56 - x^65 - x^85 + ...
G.f. = q - q^4 - q^16 + q^25 + q^49 - q^64 - q^100 + q^121 + q^169 - q^196 + ...

Mathematica

a[ n_] := SeriesCoefficient[ (EllipticTheta[ 4, 0, x^3] - EllipticTheta[ 4, 0, x^(1/3)]) / (2 x^(1/3)), {x, 0, n}]; (* Michael Somos, Jun 30 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ x, x^2] EllipticTheta[ 2, 0, x^(3/2)] / (2 x^(3/8)), {x, 0, n}]; (* Michael Somos, Jun 30 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ x, x^2] QPochhammer[ -x^3, x^3] QPochhammer[ x^6], {x, 0, n}]; (* Michael Somos, Jun 30 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ x, x^6] QPochhammer[ x^5, x^6] QPochhammer[ x^6], {x, 0, n}]; (* Michael Somos, Jun 30 2015 *)
a[ n_] := (-1)^n Sign @ SquaresR[ 1, 3 n + 1]; (* Michael Somos, Jun 30 2015 *)

Prog

(PARI) {a(n) = (-1)^n * issquare(3*n + 1)}; /* Michael Somos, Apr 12 2005 */

Crossrefs

Cf. A001082, A002448, A089801, A089812.

Sequence in context: A179776 A360130 A353570 * A089801 A290739 A143064

Adjacent sequences: A089799 A089800 A089801 * A089803 A089804 A089805

Keyword

sign

Author

Eric W. Weisstein, Nov 12 2003

Extensions

Corrected by N. J. A. Sloane, Nov 05 2005

Status

approved