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A164273
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Expansion of phi(-q) * phi(q^3) in powers of q where phi() is a Ramanujan theta function.
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4
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1, -2, 0, 2, -2, 0, 0, 4, 0, -2, 0, 0, -2, -4, 0, 0, 6, 0, 0, 4, 0, -4, 0, 0, 0, -2, 0, 2, -4, 0, 0, 4, 0, 0, 0, 0, -2, -4, 0, 4, 0, 0, 0, 4, 0, 0, 0, 0, 6, -6, 0, 0, -4, 0, 0, 0, 0, -4, 0, 0, 0, -4, 0, 4, 6, 0, 0, 4, 0, 0, 0, 0, 0, -4, 0, 2, -4, 0, 0, 4, 0, -2, 0, 0, -4, 0, 0, 0, 0, 0, 0, 8, 0, -4, 0, 0, 0, -4, 0, 0, -2, 0, 0, 4, 0
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OFFSET
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0,2
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COMMENTS
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LINKS
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FORMULA
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Expansion of eta(q)^2 * eta(q^6)^5 / (eta(q^2) * eta(q^3)^2 * eta(q^12)^2) in powers of q.
Euler transform of period 12 sequence [ -2, -1, 0, -1, -2, -4, -2, -1, 0, -1, -2, -2, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (48 t)) = 768^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A112605.
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EXAMPLE
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G.f. = 1 - 2*q + 2*q^3 - 2*q^4 + 4*q^7 - 2*q^9 - 2*q^12 - 4*q^13 + 6*q^16 + ...
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MATHEMATICA
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a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q] EllipticTheta[ 3, 0 , q^3], {q, 0, n}]; (* Michael Somos, Sep 02 2015 *)
f[x_, y_] := QPochhammer[-x, x*y]*QPochhammer[-y, x*y]*QPochhammer[x*y, x*y]; A164273[n_] := SeriesCoefficient[f[-q, -q]*f[q^3, q^3], {q, 0, n}]; Table[A164273[n], {n, 0, 50}] (* G. C. Greubel, Sep 16 2017 *)
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PROG
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(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^6 + A)^5 / (eta(x^2 + A) * eta(x^3 + A)^2 * eta(x^12 + A)^2), n))};
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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