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A216060
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Expansion of (phi(q) / phi(q^4))^2 in powers of q where phi() is a Ramanujan theta function.
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2
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1, 4, 4, 0, 0, -8, -16, 0, 0, 20, 56, 0, 0, -40, -160, 0, 0, 72, 404, 0, 0, -128, -944, 0, 0, 220, 2072, 0, 0, -360, -4320, 0, 0, 576, 8648, 0, 0, -904, -16720, 0, 0, 1384, 31360, 0, 0, -2088, -57312, 0, 0, 3108, 102364, 0, 0, -4552, -179104, 0, 0, 6592
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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)^5 * eta(q^16)^2 / (eta(q)^2 * eta(q^8)^5))^2 in powers of q.
Euler transform of period 16 sequence [ 4, -6, 4, -6, 4, -6, 4, 4, 4, -6, 4, -6, 4, -6, 4, 0, ...].
a(4*n) = 0 unless n=0. a(4*n + 3) = 0. a(4*n + 1) = 4 * A079006(n). a(4*n + 2) = 4 * A001938(n).
Empirical: Sum{n>=0} a(n)/exp(Pi*n) = 40 + 28*sqrt(2) - 8*sqrt(48+34*sqrt(2)). - Simon Plouffe, Mar 02 2021
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EXAMPLE
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1 + 4*q + 4*q^2 - 8*q^5 - 16*q^6 + 20*q^9 + 56*q^10 - 40*q^13 - 160*q^14 + ...
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MATHEMATICA
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a[n_]:= SeriesCoefficient[(EllipticTheta[3, 0, q]/EllipticTheta[3, 0, q^4])^2, {q, 0, n}]; Table[a[n], {n, 0, 50}] (* G. C. Greubel, Dec 04 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^2 + A)^5 * eta(x^16 + A)^2 / (eta(x + A)^2 * eta(x^8 + 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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