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A260162
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Expansion of phi(-x) / psi(-x^3) in powers of x where psi(), phi() are Ramanujan theta functions.
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1
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1, -2, 0, 1, 0, 0, 1, 0, 0, 0, -2, 0, 1, -2, 0, 2, 0, 0, 1, 0, 0, 1, -2, 0, 2, -4, 0, 3, -2, 0, 2, 0, 0, 1, -4, 0, 4, -6, 0, 5, -2, 0, 3, 0, 0, 3, -6, 0, 6, -10, 0, 8, -4, 0, 5, -2, 0, 4, -10, 0, 9, -14, 0, 12, -6, 0, 8, -2, 0, 7, -14, 0, 14, -22, 0, 18, -10
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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 q^(3/8) * eta(q)^2 * eta(q^6) / (eta(q^2) * eta(q^3) * eta(q^12)) in powers of q.
Euler transform of period 12 sequence [ -2, -1, -1, -1, -2, -1, -2, -1, -1, -1, -2, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (768 t)) = 12^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A132217.
G.f.: 1 / (Product_{k>0} (1 + x^k) * (1 + x^k + x^(2*k)) * (1 + x^(6*k))).
2 * a(n) = A253243(2*n + 3). a(3*n + 2) = 0.
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EXAMPLE
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G.f. = 1 - 2*x + x^3 + x^6 - 2*x^10 + x^12 - 2*x^13 + 2*x^15 + x^18 + x^21 + ...
G.f. = 1/q^3 - 2*q^5 + q^21 + q^45 - 2*q^77 + q^93 - 2*q^101 + 2*q^117 + ...
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MATHEMATICA
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a[ n_] := SeriesCoefficient[ 2^(1/2) x^(3/8) EllipticTheta[ 4, 0, x] / EllipticTheta[ 2, Pi/4, x^(3/2)], {x, 0, n}];
a[n_]:= SeriesCoefficient[EllipticTheta[3, 0, -q]*QPochhammer[-q^3, q^6]/ QPochhammer[q^6, q^6], {q, 0, n}]; Table[a[n], {n, 0, 50}] (* G. C. Greubel, Mar 19 2018 *)
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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) / (eta(x^2 + A) * eta(x^3 + A) * eta(x^12 + A)), n))};
(PARI) q='q+O('q^99); Vec(eta(q)^2*eta(q^6)/(eta(q^2)*eta(q^3)*eta(q^12))) \\ Altug Alkan, Mar 20 2018
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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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