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A115979 Expansion of (1 - theta_4(q)*theta_4(q^3))/2 in powers of q.

%I A115979 #20 May 09 2019 22:04:36
%S A115979 1,0,1,-3,0,0,2,0,1,0,0,-3,2,0,0,-3,0,0,2,0,2,0,0,0,1,0,1,-6,0,0,2,0,
%T A115979 0,0,0,-3,2,0,2,0,0,0,2,0,0,0,0,-3,3,0,0,-6,0,0,0,0,2,0,0,0,2,0,2,-3,
%U A115979 0,0,2,0,0,0,0,0,2,0,1,-6,0,0,2,0,1,0,0,-6,0,0,0,0,0,0,4,0,2,0,0,0,2,0,0,-3,0,0,2,0,0
%N A115979 Expansion of (1 - theta_4(q)*theta_4(q^3))/2 in powers of q.
%H A115979 Antti Karttunen, <a href="/A115979/b115979.txt">Table of n, a(n) for n = 1..65537</a>
%F A115979 Expansion of (1-(eta(q)*eta(q^3))^2/(eta(q^2)*eta(q^6)))/2 in powers of q.
%F A115979 Moebius transform is period 12 sequence [1,-1,0,-3,-1,0,1,3,0,1,-1,0,...].
%F A115979 a(n) is multiplicative and a(2^e) = -3(1+(-1)^e)/2 if e>0, a(3^e)=1, a(p^e) = 1+e if p == 1 (mod 6), a(p^e) = (1+(-1)^e)/2 if p == 5 (mod 6).
%F A115979 G.f.: Sum_{k>0} x^(k)/(1+x^k+x^(2k)) -4x^(4k)/(1+x^(4k)+x^(8k)).
%F A115979 a(n) = -(-1)^n*A096936(n).
%F A115979 A115978(n) = -2*a(n) if n > 0.
%p A115979 S:=series((1-JacobiTheta4(0,q)*JacobiTheta4(0,q^3))/2, q, 106):
%p A115979 seq(coeff(S,q,n),n=1..105); # _Robert Israel_, Nov 20 2017
%t A115979 Drop[CoefficientList[Series[(1 -EllipticTheta[4, 0, q]*EllipticTheta[4, 0, q^3])/2, {q, 0, 110}], q], 1] (* _G. C. Greubel_, May 09 2019 *)
%o A115979 (PARI) {a(n)=local(A); if(n<1, 0, A=x*O(x^n); polcoeff( (eta(x+A)*eta(x^3+A))^2/eta(x^2+A)/eta(x^6+A), n)/-2)}
%o A115979 (Scheme) (define (A115979 n) (- (* (expt -1 n) (A096936 n)))) ;; Follow A096936 for the rest of code. - _Antti Karttunen_, Nov 20 2017
%o A115979 (Sage)
%o A115979 def E(x): return 1 + 2*sum((-1)^k*x^(k^2) for k in (1..50))
%o A115979 a=((1 - E(x)*E(x^3))/2).series(x, 110).coefficients(x, sparse=False); a[1:] # _G. C. Greubel_, May 09 2019
%Y A115979 Cf. A033716, A096936, A115978.
%K A115979 sign,mult
%O A115979 1,4
%A A115979 _Michael Somos_, Feb 09 2006