A248394 q-Expansion of the modular form of weight 3/2, g*theta(2) in Tunnell's notation (see Comments).

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

Offset: 0

Author

N. J. A. Sloane, Oct 18 2014

Keywords

• sign

Comments

g = q*Product_{m=1..oo} (1-q^(8*m))*(1-q^(16*m)),

theta(t) = Sum_{n=-oo..oo} q^(t*n^2).

Although the OEIS does not normally include sequences in which every other term is zero, this one is important enough to warrant an exception.

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..10000
• 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, Jacobi Theta Functions

Crossrefs

The nonzero bisection is A034950, which has further information and references.

Used in A248397-A248406.

Cf. A000122 (theta_3(q)), A072068, A072069, A080917, A080918, A248395.

Programs

Maple

# This produces a list of the first 100 terms:
g:=q*mul((1-q^(8*m))*(1-q^(16*m)),m=1..30);
g:=series(g,q,100);
th:=t->series( add(q^(t*n^2),n=-50..50), q, 100);
series(g*th(2),q,100);
seriestolist(%);
# Alternative with https://oeis.org/transforms.txt and the Somos Euler transform in A034950:
p8 := [2,-3,2,-2,2,-3,2,-3] ;
L := [seq(op(p8),i=1..10)] ;
EULER(%) ;
[1,op(%)] ;
[0,op(AERATE(%,1))] ; # R. J. Mathar, Nov 11 2014

Mathematica

QP = QPochhammer; s = q*QP[q^8]*QP[q^16]*EllipticTheta[3, 0, q^2] + O[q]^80; CoefficientList[s, q] (* Jean-François Alcover, Nov 27 2015 *)

Formula

From Seiichi Manyama, Sep 30 2018: (Start)

Let q = exp(Pi i t).

theta_3(q) = 1 + 2*q + 2*q^4 + 2*q^9 + 2*q^16 + ... .

G.f.: (theta_3(q) - theta_3(q^4))*(theta_3(q^32) - theta_3(q^8)/2)*theta_3(q^2).

a(2*n-1) = A080918(2*n-1) - A080917(2*n-1)/2 = A072069(n) - A072068(n)/2 for n > 0. (End)