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A115660
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Expansion of (phi(q) * phi(q^6) - phi(q^2) * phi(q^3)) / 2 in powers of q where phi() is a Ramanujan theta function.
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12
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1, -1, -1, 1, -2, 1, 2, -1, 1, 2, -2, -1, 0, -2, 2, 1, 0, -1, 0, -2, -2, 2, 0, 1, 3, 0, -1, 2, -2, -2, 2, -1, 2, 0, -4, 1, 0, 0, 0, 2, 0, 2, 0, -2, -2, 0, 0, -1, 3, -3, 0, 0, -2, 1, 4, -2, 0, 2, -2, 2, 0, -2, 2, 1, 0, -2, 0, 0, 0, 4, 0, -1, 2, 0, -3, 0, -4, 0
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OFFSET
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1,5
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COMMENTS
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Number 41 of the 74 eta-quotients listed in Table I of Martin (1996). - Michael Somos, Mar 14 2012
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LINKS
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FORMULA
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Expansion of eta(q) * eta(q^4) * eta(q^6) * eta(q^24) / (eta(q^3) * eta(q^8)) in powers of q.
Euler transform of period 24 sequence [ -1, -1, 0, -2, -1, -1, -1, -1, 0, -1, -1, -2, -1, -1, 0, -1, -1, -1, -1, -2, 0, -1, -1, -2, ...].
a(n) is multiplicative with a(2^e) = a(3^e) = (-1)^e, a(p^e) = e+1 if p == 1, 7 (mod 24), a(p^e) = (e+1) * (-1)^e if p == 5, 11 (mod 24), a(p^e) = (1 + (-1)^e) / 2 if p == 13, 17, 19, 23 (mod 24).
G.f. is a period 1 Fourier series which satisfies f(-1 / (24 t)) = 24^(1/2) (t/i) f(t) where q = exp(2 Pi i t).
G.f.: Sum_{k>0} Kronecker(k,8) * x^k / (1 + x^k + x^(2*k)) = Sum_{k>0} Kronecker(k,3) * x^k * (1 - x^(2*k)) / (1 + x^(4*k)).
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EXAMPLE
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G.f. = q - q^2 - q^3 + q^4 - 2*q^5 + q^6 + 2*q^7 - q^8 + q^9 + 2*q^10 - 2*q^11 + ...
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MATHEMATICA
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a[ n_] := SeriesCoefficient[ q QPochhammer[ q] QPochhammer[ q^4] QPochhammer[ q^6] QPochhammer[ q^24] / (QPochhammer[ q^3] QPochhammer[ q^8]), {q, 0, n}]; (* Michael Somos, Apr 19 2015 *)
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q] EllipticTheta[ 3, 0, q^6] - EllipticTheta[ 3, 0, q^2] EllipticTheta[ 3, 0, q^3]) / 2, {q, 0, n}]; (* Michael Somos, Apr 19 2015 *)
a[ n_] := If[ n < 1, 0, Sum[ KroneckerSymbol[ 2, d] KroneckerSymbol[ -3, n/d], {d, Divisors[ n]}]]; (* Michael Somos, Apr 19 2015 *)
a[ n_] := If[ n < 1, 0, Times @@ (Which[ # == 1, 1, # < 5, (-1)^#2, Mod[#, 24] < 12, (#2 + 1) KroneckerSymbol[ #, 12]^#2, True, 1 - Mod[#2, 2]]& @@@ FactorInteger[n])]; (* Michael Somos, Oct 22 2015 *)
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PROG
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(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<5, (-1)^e, p%24<12, (e+1) * kronecker( p, 12)^e, 1-e%2)))};
(PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x + A) * eta(x^4 + A) * eta(x^6 + A) * eta(x^24 + A) / (eta(x^3 + A) * eta(x^8 + A)), n))};
(PARI) {a(n) = if( n<1, 0, sumdiv( n, d, kronecker( 2, d) * kronecker( -3, n/d)))};
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CROSSREFS
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Cf. A000377, A046113, A108563, A128580, A128581, A192013, A259668, A260308, A261115, A263548, A263571.
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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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