A159814 Expansion of eta(z)^2*eta(4*z)^6/eta(2*z).

1, -2, 0, 0, -4, 12, 0, 0, -3, -20, 0, 0, 28, -8, 0, 0, -8, 42, 0, 0, -72, -20, 0, 0, 29, 36, 0, 0, 84, -72, 0, 0, 24, -40, 0, 0, -68, 36, 0, 0, -112, 24, 0, 0, 84, 248, 0, 0, -39, -158, 0, 0, -12, -144, 0, 0, 216, -116, 0, 0, -108, -16, 0, 0, 80, 144, 0, 0, 48, 152, 0, 0, -232, 220

Offset

1,2

Comments

Expansion of eta(q)^2*eta(q^4)^6/eta(q^2) in powers of q. Unique cusp form of weight 7/2, level 8 and trivial character.

Links

G. C. Greubel, Table of n, a(n) for n = 1..1000

Formula

Euler transform of period 4 sequence [ -2, -1, -2, -7, ...]. - Michael Somos, Jun 07 2012

a(4*n) = a(4*n + 3) = 0. - Michael Somos, Jun 07 2012

Example

q - 2*q^2 - 4*q^5 + 12*q^6 - 3*q^9 - 20*q^10 + 28*q^13 - 8*q^14 - 8*q^17 + ...

Mathematica

max = 80; a = Table[{-2, -1, -2, -7}, {max/4}] // Flatten; Series[Product[1/(1 - x^n)^a[[n]], {n, 1, max}], {x, 0, max}] // CoefficientList[#, x]& (* Jean-François Alcover, Jun 10 2013, after Michael Somos *)
eta[q_]:= q^(1/24)*QPochhammer[q];  A159814:= CoefficientList[Series[ eta[q]^2*eta[q^4]^6/eta[q^2], {q, 0, 100}], q]; Table[A159814[[n]], {n, 2, 100}] (* G. C. Greubel, May 19 2018 *)

Prog

(Magma) Basis(CuspidalSubspace(HalfIntegralWeightForms(8, 7/2)), 100)
(PARI) {a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^4 + A)^6 / eta(x^2 + A), n))} /* Michael Somos, Jun 07 2012 */
(PARI) q='q+O('q^99); Vec(eta(q)^2*eta(q^4)^6/eta(q^2)) \\ Joerg Arndt, May 21 2018

Crossrefs

Sequence in context: A011449 A231037 A048243 * A169774 A302689 A289088

Adjacent sequences: A159811 A159812 A159813 * A159815 A159816 A159817

Keyword

sign

Author

Steven Finch, Apr 22 2009

Status

approved