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A210067
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Expansion of (phi(-q) / phi(q^2))^2 in powers of q where phi() is a Ramanujan theta function.
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3
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1, -4, 0, 16, 0, -56, 0, 160, 0, -404, 0, 944, 0, -2072, 0, 4320, 0, -8648, 0, 16720, 0, -31360, 0, 57312, 0, -102364, 0, 179104, 0, -307672, 0, 519808, 0, -864960, 0, 1419456, 0, -2299832, 0, 3682400, 0, -5831784, 0, 9141808, 0, -14194200, 0, 21842368, 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^2) * eta(q^8)^2 / eta(q^4)^5)^2 in powers of q.
Euler transform of period 8 sequence [ -4, -6, -4, 4, -4, -6, -4, 0, ...].
Empirical: Sum_{n>=0} a(n)/exp(2*Pi*n) = -32 - 24*sqrt(2) + 4*sqrt(140+99*sqrt(2)). - Simon Plouffe, Mar 02 2021
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EXAMPLE
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1 - 4*q + 16*q^3 - 56*q^5 + 160*q^7 - 404*q^9 + 944*q^11 - 2072*q^13 + ...
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MATHEMATICA
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a[n_] := SeriesCoefficient[(EllipticTheta[3, 0, -q]/EllipticTheta[3, 0, q^2])^2, {q, 0, n}]; Table[a[n], {n, 0, 50}] (* G. C. Greubel, Nov 29 2017 *)
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PROG
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(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A)^2 * eta(x^2 + A) * eta(x^8 + A)^2 / eta(x^4 + A)^5)^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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