A248395 q-Expansion of the modular form of weight 3/2, g*theta(4) in Tunnell's notation (see Comments).

0, 1, 0, 0, 0, 2, 0, 0, 0, -1, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, -4, 0, 0, 0, -1, 0, 0, 0, 2, 0, 0, 0, -4, 0, 0, 0, 2, 0, 0, 0, 4, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, -2, 0, 0, 0, 4, 0, 0, 0, 2, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, -4, 0, 0, 0, 0

Offset

0,6

Comments

g = Product_{m>=1} ((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 only every fourth term is nonzero, this one is important enough to warrant an exception.

Links

G. C. Greubel, Table of n, a(n) for n = 0..5000

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

Formula

From G. C. Greubel, Jul 02 2018: (Start)

Expansion of eta(q^8)*eta(q^16)*theta_{3}(0, q^4)/q in powers of q.

Expansion of eta(q^8)^6/(q*eta(q^4)^2*eta(q^16)). (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(4), q, 100);
seriestolist(%);

Mathematica

QP := QPochhammer; a:= CoefficientList[Series[QP[q^8]*QP[q^16]* EllipticTheta[3, 0, q^4], {q, 0, 60}], q]; Join[{0}, Table[a[[n]], {n, 1, 50}]] (* G. C. Greubel, Jul 02 2018 *)

Prog

(PARI) q='q+O('q^50); A = eta(q^8)^6/(q*eta(q^4)^2*eta(q^16)); concat([0], Vec(A)) \\ G. C. Greubel, Jul 02 2018

Crossrefs

The nonzero quadrisection is A080966, which has further information and references.

Cf. A248394.

Used in A248397-A248406.

Sequence in context: A374247 A369165 A056170 * A059483 A067618 A279255

Adjacent sequences: A248392 A248393 A248394 * A248396 A248397 A248398

Keyword

sign

Author

N. J. A. Sloane, Oct 18 2014

Status

approved