A109039 Expansion of eta(q) * eta(q^3) * (eta(q^4) * eta(q^6) / eta(q^12))^2 in powers of q.

1, -1, -1, -1, -1, 4, -1, 6, -1, -1, 4, -12, -1, -14, 6, 4, -1, 16, -1, 18, 4, 6, -12, -24, -1, -21, -14, -1, 6, 28, 4, 30, -1, -12, 16, -24, -1, -38, 18, -14, 4, 40, 6, 42, -12, 4, -24, -48, -1, -43, -21, 16, -14, 52, -1, 48, 6, 18, 28, -60, 4, -62, 30, 6

Offset

0,6

Comments

Number 25 of the 74 eta-quotients listed in Table I of Martin (1996).

Links

Table of n, a(n) for n=0..63.

Yves Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.

Michael Somos, Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers, 2016.

Formula

Euler transform of period 12 sequence [ -1, -1, -2, -3, -1, -4, -1, -3, -2, -1, -1, -4, ...].

G.f.: Product_{k>0} (1 - x^k) * (1 - x^(3*k)) * (1 - x^(4*k))^2 / (1 + x^(6*k))^2.

a(n) = -A109040(n) unless n=0. a(2*n) = a(3*n) = a(n).

G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 12^(3/2) (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A124815. - Michael Somos, May 18 2015

Sum_{k=1..n} abs(a(k)) ~ c * n^2, where c = Pi^2/(24*sqrt(3)) = 0.237425... . - Amiram Eldar, Jan 29 2024

Example

G.f. = 1 - q - q^2 - q^3 - q^4 + 4*q^5 - q^6 + 6*q^7 - q^8 - q^9 + 4*q^10 + ...

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ q] QPochhammer[ q^3] (QPochhammer[ q^4] QPochhammer[ q^6] / QPochhammer[ q^12])^2, {q, 0, n}]; (* Michael Somos, May 18 2015 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q^3] QPochhammer[ q^3, q^6]^3 EllipticTheta[ 2, 0, q^(1/2)] EllipticTheta[ 2, Pi/4, q^(1/2)]^2 / (4 q^(3/8)), {q, 0, n}]; (* Michael Somos, May 18 2015 *)

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^3 + A) * eta(x^4 + A)^2 * eta(x^6 + A)^2 / eta(x^12 + A)^2, n))};
(Magma) A := Basis( ModularForms( Gamma1(12), 2), 64); A[1] - A[2] - A[3] - A[4] - A[5] + 4*A[6] - A[7] + 6*A[8] - A[9]; /* Michael Somos, May 18 2015 */

Crossrefs

Cf. A109040, A124815.

Sequence in context: A010642 A127168 A176216 * A257656 A109040 A133828

Adjacent sequences: A109036 A109037 A109038 * A109040 A109041 A109042

Keyword

sign

Author

Michael Somos, Jun 17 2005

Status

approved