A034950 Expansion of eta(8z)*eta(16z)*theta_3(2z).

1, 2, 0, 0, 1, -2, 0, 0, -4, -2, 0, 0, -3, 0, 0, 0, 4, -4, 0, 0, 0, 6, 0, 0, 1, 4, 0, 0, 4, 2, 0, 0, 0, -2, 0, 0, 4, -2, 0, 0, -3, 2, 0, 0, -4, -4, 0, 0, -4, 2, 0, 0, -8, -6, 0, 0, 8, -4, 0, 0, 1, -4, 0, 0, -4, 6, 0, 0, 0, 2, 0, 0, 0, -2, 0, 0, 4, 8, 0, 0, 0, 6, 0, 0, 5, -2, 0, 0, 4, -2, 0, 0, 8, 4, 0, 0, -4, -8, 0, 0, -4, 8, 0, 0, 4

Offset

0,2

Comments

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

Links

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from G. C. Greubel)

Ken Ono and Christopher Skinner, Fourier Coefficients of Half-Integral Weight Modular Forms Modulo l, Ann. Math., 147 (1998), 453-470.

Michael Somos, Introduction to Ramanujan theta functions

J. B. Tunnell, A classical Diophantine problem and modular forms of weight 3/2, Invent. Math., 72 (1983), 323-334.

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Euler transform of period 8 sequence [2, -3, 2, -2, 2, -3, 2, -3, ...]. - Michael Somos, Feb 16 2006

Expansion of q^(-1/2) * eta(q^2)^5 * eta(q^8) / (eta(q)^2 * eta(q^4)) in powers of q. - Michael Somos, Feb 16 2006

Expansion of psi(x)^2 * psi(-x^2) = phi(x) * psi(x^2) * psi(-x^2) = phi(x) * psi(x^4) * phi(-x^4) in powers of x where phi(), psi() are Ramanujan theta functions. - Michael Somos, Feb 18 2015

G.f.: Product_{k>0} (1 + x^k)^2 * (1 - x^(2*k))^3 * (1 + x^(4*k)). - Michael Somos, Feb 16 2006

2 * a(n) = A080963(2*n + 1). a(4*n + 2) = a(4*n + 3) = 0. - Michael Somos, Feb 18 2015

a(n) = A072069(n+1) - A072068(n+1)/2. - _Seichi Manymama_, Sep 30 2018

Example

G.f. = 1 + 2*x + x^4 - 2*x^5 - 4*x^8 - 2*x^9 - 3*x^12 + 4*x^16 - 4*x^17 + ...
G.f. = q + 2*q^3 + q^9 - 2*q^11 - 4*q^17 - 2*q^19 - 3*q^25 + 4*q^33 - ...

Mathematica

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x] EllipticTheta[ 2, 0, x] EllipticTheta[ 2, Pi/4, x] / Sqrt[8 x], {x, 0, n}]; (* Michael Somos, Feb 18 2015 *)
QP = QPochhammer; s = QP[q^2]^5*(QP[q^8]/(QP[q]^2*QP[q^4])) + O[q]^105; CoefficientList[s, q] (* Jean-François Alcover, Nov 27 2015 *)

Prog

(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^5 * eta(x^8 + A) / (eta(x + A)^2 * eta(x^4 + A)), n))}; /* Michael Somos, Feb 16 2006 */

Crossrefs

Cf. A072068, A072069, A080963.

A bisection of A248394.

Sequence in context: A143380 A143377 A367116 * A351127 A331816 A099584

Adjacent sequences: A034947 A034948 A034949 * A034951 A034952 A034953

Keyword

sign

Author

N. J. A. Sloane

Status

approved