A286137 Expansion of eta(q) * eta(q^2) * eta(q^15) * eta(q^30) in powers of q.

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

Offset

0,5

Links

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Michael Somos, A Remarkable eta-product Identity

Formula

G.f.: x^2 * Prod_{k>0} (1 - x^k) * (1 - x^(2 * k)) * (1 - x^(15 * k)) * (1 - x^(30 * k)).

G.f. is a period 1 Fourier series which satisfies f(-1 / (30 t)) = 30 (t/i)^2 f(t) where q = exp(2 Pi i t). - Michael Somos, Nov 26 2019

a(3*n) = -A030218(n). - Michael Somos, Mar 10 2020

Example

G.f. = x^2 - x^3 - 2*x^4 + x^5 + 2*x^7 + x^8 - 2*x^11 + x^12 + ... - Michael Somos, Mar 10 2020

Maple

seq(coeff(series(x^2*mul((1-x^k)*(1-x^(2*k))*(1-x^(15*k))*(1-x^(30*k)), k=1..n), x, n+1), x, n), n=0..150); # Muniru A Asiru, Jul 29 2018

Mathematica

eta[q_] := q^(1/24)*QPochhammer[q]; CoefficientList[Series[eta[q] *eta[q^2]*eta[q^15]*eta[q^30], {q, 0, 50}], q] (* G. C. Greubel, Jul 29 2018 *)

Prog

(PARI) q='q+O('q^50); A = eta(q)*eta(q^2)*eta(q^15)*eta(q^30); concat([0, 0], Vec(A)) \\ G. C. Greubel, Jul 29 2018
(Magma) A := Basis( CuspForms( Gamma0(30), 2), 80); A[2] - A[3]; /* Michael Somos, Nov 26 2019 */

Crossrefs

Cf. A030218, A122776.

Sequence in context: A030204 A083650 A138514 * A320658 A284966 A143540

Adjacent sequences: A286134 A286135 A286136 * A286138 A286139 A286140

Keyword

sign

Author

Seiichi Manyama, May 03 2017

Status

approved