A089810 Expansion of Jacobi theta function theta_3(Pi/6, q) in powers of q.

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

Offset

0,10

Comments

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

This is an example of the quintuple product identity in the form f(a*b^4, a^2/b) - (a/b) * f(a^4*b, b^2/a) = f(-a*b, -a^2*b^2) * f(-a/b, -b^2) / f(a, b) where a = -x^5, b = -x. - Michael Somos, Jul 12 2012

Convolution square is A258279. - Michael Somos, May 25 2015

Number 8 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, 2019.

Eric Weisstein's World of Mathematics, Jacobi Theta Functions.

Eric Weisstein's World of Mathematics, Quintuple Product Identity.

Eric Weisstein's World of Mathematics, Ramanujan 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 Jacobi theta function (3theta_4(q^9) - theta_4(q)) / 2 in powers of q.

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

From Michael Somos, Nov 05 2005: (Start)

Euler transform of period 6 sequence [ 1, -1, 0, -1, 1, -1, ...].

G.f.: (Sum_{k in Z} 3 * (-x)^((3*k)^2) - (-x)^(k^2)) / 2 = Product_{k>0} (1 - x^(2*k)) / ((1 - x^(6*k - 1)) * (1 - x^(6*k-5))).

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

Expansion of psi(q) * chi(-q^3) in powers of q where psi(), chi() are Ramanujan theta functions. - Michael Somos, Sep 16 2007

Expansion of (3 * phi(-q^9) - phi(-q)) / 2 in powers of q where phi() is a Ramanujan theta function.

From Michael Somos, Sep 17 2007: (Start)

Expansion of Jacobi theta function theta_3(Pi/6, q) in powers of q.

Expansion of f(x*w, x/w) in powers of x where w is a primitive sixth root of unity and f() is Ramanujan's two-variable theta function. (End)

From Michael Somos, Jan 26 2008: (Start)

G.f. is a period 1 Fourier series which satisfies f(-1 / (144 t)) = 72^(1/2) (t/i)^(1/2) g(t) where q = exp(2 Pi i t) and g(t) is the g.f. for A080995.

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

a(3*n + 2) = a(4*n + 2) = a(4*n + 3) = a(5*n + 2) = a(5*n + 3) = a(8*n + 5) = a(9*n + 3) = a(9*n + 6) = 0. a(3*n + 1) = A089802(n). a(4*n) = A089807(n). a(9*n) = A002448(n).

a(n) = (floor(sqrt(n))-floor(sqrt(n-1)))*(abs(2-4*sin((floor(sqrt(n))+1)*Pi/3)^2) - 4*sin((floor(sqrt(n))+2)*Pi/3)^2)*(-1)^floor(floor(sqrt(n)-1)/3). - Mikael Aaltonen, Jan 17 2015

From Michael Somos, May 25 2015: (Start)

a(n) = (-1)^n * A089807(n) = A204843(4*n) = A204853(4*n).

a(8*n + 1) = A089812(n). a(12*n + 4) = - A089801(n). (End)

Sum_{k=1..n} abs(a(k)) ~ (4/3)*sqrt(n). - Amiram Eldar, Jan 27 2024

Example

G.f. = 1 + q - q^4 - 2*q^9 - q^16 + q^25 + 2*q^36 + q^49 - q^64 - 2*q^81 + ...

Mathematica

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, Pi/6, q], {q, 0, n}]; (* Michael Somos, Nov 14 2011 *)
a[ n_] := SeriesCoefficient[ (3 EllipticTheta[ 4, 0, q^9] - EllipticTheta[ 4, 0, q])  /2, {q, 0, n}]; (* Michael Somos, Nov 14 2011 *)
QP = QPochhammer; s = QP[q^2]^2*(QP[q^3] / (QP[q]*QP[q^6])) + O[q]^100; CoefficientList[s, q] (* Jean-François Alcover, Nov 09 2015, adapted from PARI *)

Prog

(PARI) {a(n) = my(x); if( n<1, n==0, issquare(n, &x) * (1 + (n%3==0)) * (-1)^((1 + x) \ 3))}; /* Michael Somos, Nov 05 2005 */
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^3 + A) / (eta(x + A) * eta(x^6 + A)), n))}; /* Michael Somos, Jan 26 2008 */

Crossrefs

Cf. A002448, A080995, A089801, A089802, A089807, A089812, A204843, A204853, A258279.

Cf. A000700, A000122, A010054, A121373.

Sequence in context: A280618 A347714 A089807 * A214411 A324179 A216577

Adjacent sequences: A089807 A089808 A089809 * A089811 A089812 A089813

Keyword

sign,mult

Author

Eric W. Weisstein, Nov 12 2003

Status

approved