A159814 - OEIS

%I #26 Sep 08 2022 08:45:44

%S 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,

%T 0,0,84,-72,0,0,24,-40,0,0,-68,36,0,0,-112,24,0,0,84,248,0,0,-39,-158,

%U 0,0,-12,-144,0,0,216,-116,0,0,-108,-16,0,0,80,144,0,0,48,152,0,0,-232,220

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

%C 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.

%H G. C. Greubel, <a href="/A159814/b159814.txt">Table of n, a(n) for n = 1..1000</a>

%F Euler transform of period 4 sequence [ -2, -1, -2, -7, ...]. - _Michael Somos_, Jun 07 2012

%F a(4*n) = a(4*n + 3) = 0. - _Michael Somos_, Jun 07 2012

%e 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 + ...

%t 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_ *)

%t 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 *)

%o (Magma) Basis(CuspidalSubspace(HalfIntegralWeightForms(8,7/2)),100)

%o (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 */

%o (PARI) q='q+O('q^99); Vec(eta(q)^2*eta(q^4)^6/eta(q^2)) \\ _Joerg Arndt_, May 21 2018

%K sign

%O 1,2

%A _Steven Finch_, Apr 22 2009