login

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).

A159814
Expansion of eta(z)^2*eta(4*z)^6/eta(2*z).
1
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
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
KEYWORD
sign
AUTHOR
Steven Finch, Apr 22 2009
STATUS
approved