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A138746
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Expansion of 1 - eta(q) * eta(q^3) * eta(q^4)^3 / (eta(q^2)^2 * eta(q^12)) in powers of q.
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4
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1, -1, 3, -1, 2, -3, 0, -1, 1, -2, 0, -3, 2, 0, 6, -1, 2, -1, 0, -2, 0, 0, 0, -3, 3, -2, 3, 0, 2, -6, 0, -1, 0, -2, 0, -1, 2, 0, 6, -2, 2, 0, 0, 0, 2, 0, 0, -3, 1, -3, 6, -2, 2, -3, 0, 0, 0, -2, 0, -6, 2, 0, 0, -1, 4, 0, 0, -2, 0, 0, 0, -1, 2, -2, 9, 0, 0, -6
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OFFSET
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1,3
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COMMENTS
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LINKS
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FORMULA
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Expansion of 1 - psi(-q) * psi(q^2) * chi(-q^3) * chi(-q^6) in powers of q where psi(), chi() are Ramanujan theta functions.
Moebius transform is period 24 sequence [ 1, -2, 2, 0, 1, -4, -1, 0, -2, -2, -1, 0, 1, 2, 2, 0, 1, 4, -1, 0, -2, 2, -1, 0, ...].
a(n) is multiplicative with a(2^e) = -1 if e>0, a(3^e) = 2 - (-1)^e, a(p^e) = e+1 if p == 1, 5 (mod 12), a(p^e) = (1 + (-1)^e) / 2 if p == 7, 11 (mod 12).
G.f.: Sum_{k>0} -(-1)^k * ( f(6*k - 1) + 2 * f(6*k - 3) + f(6*k - 5) ) where f(k) := x^k / (1 + x^k).
a(12*n + 7) = a(12*n + 11) = 0.
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EXAMPLE
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G.f = q - q^2 + 3*q^3 - q^4 + 2*q^5 - 3*q^6 - q^8 + q^9 - 2*q^10 - 3*q^12 + ...
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MATHEMATICA
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a[ n_] := If[ n < 1, 0, -(-1)^n DivisorSum[ n, (-1)^Quotient[#, 6] {1, 0, 2, 0, 1, 0}[[Mod[#, 6, 1]]] &]]; (* Michael Somos, Sep 08 2015 *)
a[ n_] := If[ n < 1, 0, Times @@ (Which[ # < 3, -(-1)^#, # == 3, Mod[#2, 2] 2 + 1, Mod[#, 4] == 1, #2 + 1, True, 1 - Mod[#2, 2]] & @@@ FactorInteger @ n)]; (* Michael Somos, Sep 08 2015 *)
QP = QPochhammer; s = (1/q)*(1-QP[q]*QP[q^3]*(QP[q^4]^3/(QP[q^2]^2*QP[q^12] ))) + O[q]^80; CoefficientList[s, q] (* Jean-François Alcover, Nov 27 2015 *)
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PROG
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(PARI) {a(n) = if( n<1, 0, -(-1)^n * sumdiv(n, d, ((d%2) * ((d%3==0) + 1)) * (-1)^(d\6)))};
(PARI) {a(n) = my(A, p, e); if( n<1, 0, A = factor(n); prod(k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==2, -1, p==3, 2 - (-1)^e, p%12<6, e+1, 1-e%2)))};
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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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