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A115979
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Expansion of (1 - theta_4(q)*theta_4(q^3))/2 in powers of q.
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5
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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, 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, 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
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OFFSET
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1,4
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LINKS
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FORMULA
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Expansion of (1-(eta(q)*eta(q^3))^2/(eta(q^2)*eta(q^6)))/2 in powers of q.
Moebius transform is period 12 sequence [1,-1,0,-3,-1,0,1,3,0,1,-1,0,...].
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).
G.f.: Sum_{k>0} x^(k)/(1+x^k+x^(2k)) -4x^(4k)/(1+x^(4k)+x^(8k)).
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MAPLE
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S:=series((1-JacobiTheta4(0, q)*JacobiTheta4(0, q^3))/2, q, 106):
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MATHEMATICA
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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 *)
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PROG
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(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)}
(Sage)
def E(x): return 1 + 2*sum((-1)^k*x^(k^2) for k in (1..50))
a=((1 - E(x)*E(x^3))/2).series(x, 110).coefficients(x, sparse=False); a[1:] # G. C. Greubel, May 09 2019
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CROSSREFS
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KEYWORD
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sign,mult
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AUTHOR
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STATUS
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approved
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