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

0,4

Comments

g = 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

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)

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 *)

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.

Sequence in context: A086015 A249856 A086012 * A368843 A127268 A252459

Adjacent sequences: A248391 A248392 A248393 * A248395 A248396 A248397

Keyword

sign

Author

N. J. A. Sloane, Oct 18 2014

Status

approved