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%I #11 Sep 08 2022 08:45:45
%S 1,-1,2,-1,0,-2,3,0,0,-2,1,-2,2,0,2,-1,0,-2,0,-2,4,-1,0,0,2,0,0,-4,1,
%T -2,2,0,2,0,0,-2,3,0,0,-2,0,-4,2,0,2,-1,2,0,0,0,2,-2,0,-2,2,-3,2,-2,0,
%U 0,0,0,4,0,0,-2,3,0,0,-4
%N Expansion of q-series Sum_{n >= 0} (-1)^n*q^(n*(n + 1)/2)*(1 - q)*(1 - q^2)*...*(1 - q^n) / ((1 + q)*(1 + q^2)*(1 + q^3)*...*(1 + q^n)).
%C (-1)^n*a(n) is the number of inequivalent elements of norm 8n+1 in Z[sqrt(2)].
%H G. C. Greubel, <a href="/A161515/b161515.txt">Table of n, a(n) for n = 0..1000</a>
%H Daniel Corson, David Favero, Kate Liesinger, Sarah Zubairy, <a href="https://doi.org/10.1016/j.jnt.2004.03.002">Characters and q-series in Q(sqrt(2))</a>, J. Number Theory, 107 (2004), 392-405.
%H Jeremy Lovejoy, <a href="https://doi.org/10.1016/j.jnt.2003.12.014">Overpartitions and real quadratic fields</a>, J. Number Theory, 106 (2004), 178-186.
%t CoefficientList[Series[Sum[(-1)^n*q^(Binomial[n+1, 2])*QPochhammer[q, q]^2*QPochhammer[q^(2n+2), q^2]/(QPochhammer[q^(n+1),q]^2* QPochhammer[q^2, q^2]), {n, 0, 50}], {q, 0, 80}], q] (* _G. C. Greubel_, Dec 04 2018 *)
%o (PARI) my(q='q+O('q^80)); Vec(sum(n=0,100, (-1)^n*q^(binomial(n+1,2))* prod(k=1,n,1-q^k)/prod(j=1,n, 1+q^j))) \\ _G. C. Greubel_, Dec 04 2018
%o (Magma) m:=100; R<q>:=PowerSeriesRing(Integers(), m); [1] cat Coefficients(R!( (&+[(-1)^n*q^(Binomial(n+1,2))*(&*[1-q^k: k in [1..n]])/(&*[1+q^j:j in [1..n]]): n in [1..100]]) )); // _G. C. Greubel_, Dec 04 2018
%o (Sage)
%o from sage.combinat.q_analogues import q_pochhammer
%o s=(sum( (-1)^n*x^(binomial(n+1,2))*q_pochhammer(n,x,x)^2/q_pochhammer(n,x^2,x^2)
%o for n in range(80))).series(x, 80);
%o s.coefficients(x, sparse=False) # _G. C. Greubel_, Dec 04 2018
%K sign
%O 0,3
%A _Jeremy Lovejoy_, Jun 12 2009
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