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
KEYWORD
sign
AUTHOR
Steven Finch, Apr 22 2009
STATUS
approved