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A160806 Expansion of q^(-1/3) * (eta(q) * eta(q^7) + eta(q^4) * eta(q^28)) in powers of q^2.

%I A160806 #6 Sep 23 2018 21:02:23
%S A160806 1,-1,0,0,1,0,-2,-2,1,0,0,2,0,2,0,0,0,0,-2,0,-1,2,0,0,0,-2,0,2,1,-1,0,
%T A160806 0,-2,0,0,-2,0,0,0,0,0,0,0,2,0,0,2,0,1,0,2,0,0,0,0,-2,-2,-1,0,0,1,0,2,
%U A160806 -2,0,0,0,0,0,0,2,0,0,0,0,0,-2,2,-2,0,0,-2,0,-2,0,0,0,0,-1,0,2,2,-2,0,0,2,0,0,0,0,0,0,2,0,1
%N A160806 Expansion of q^(-1/3) * (eta(q) * eta(q^7) + eta(q^4) * eta(q^28)) in powers of q^2.
%H A160806 G. C. Greubel, <a href="/A160806/b160806.txt">Table of n, a(n) for n = 0..1000</a>
%F A160806 a(n) = b(6*n + 1) where b(n) is multiplicative and b(2^e) = b(3^e) = 0^e, b(p^e) = (-1)^e if p = 7, b(p^e) = (1+(-1)^e)/2 if p == 3, 5, 6 (mod 7), else p == 1, 2, 4 (mod 7) and p=y^2+7x^2 when b(p^2e) = (-1)^e if x*y not divisible by 3, b(p^e) = e+1 if x divisible by 3 or (e+1)(-1)^e if y divisible by 3.
%F A160806 A002655(2*n) = a(n).
%e A160806 G.f. = q - q^7 + q^25 - 2*q^37 - 2*q^43 + q^49 + 2*q^67 + 2*q^79 - 2*q^109 + ...
%t A160806 eta[q_] := q^(1/24)*QPochhammer[q]; CoefficientList[Series[q^(-1/6)*(eta[q^(1/2)]*eta[q^(7/2)] + eta[q^2]*eta[q^14]), {q, 0, 100}], q] (* _G. C. Greubel_, Sep 21 2018 *)
%o A160806 (PARI) {a(n) = local(A, p, e, x, y); if(n<0, 0, n = 6*n + 1; A = factor(n); prod(k=1, matsize(A)[1], if(p=A[k,1], e=A[k,2]; if(p==7, (-1)^e, if(kronecker(p,7)==-1, !(e%2), for(x=0,sqrtint(p\7), if(issquare(p - 7*x^2, &y), y=if(x%3&y%3, real(I^e), (e+1) * if(x%3, (-1)^e, 1)); break)); y)))))}
%o A160806 (PARI) {a(n) = local(A); if(n<0, 0, n *= 2; A = x * O(x^n); polcoeff( eta(x + A) * eta(x^7 + A), n))}
%K A160806 sign
%O A160806 0,7
%A A160806 _Michael Somos_, May 26 2009