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A245668
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Expansion of (chi(q^3) * psi(-q))^3 in powers of q where chi(), psi() are Ramanujan theta functions.
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2
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1, -3, 3, -1, -3, 6, -3, 0, 3, 3, -12, 6, -1, -12, 12, 0, -3, 12, 9, -12, 6, -6, -12, 0, -3, -15, 18, 5, 0, 18, -6, 0, 3, -6, -24, 12, 3, -12, 18, 0, -12, 24, -6, -12, 6, 18, -24, 0, -1, -27, 21, -6, -12, 18, 15, 0, 12, -6, -12, 18, 0, -36, 24, 0, -3, 24, -12
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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 phi(q^3) * psi(-q)^3 / psi(-q^3) in powers of q where phi(), psi() are Ramanujan theta functions.
Expansion of (eta(q) * eta(q^4) * eta(q^6)^2 / (eta(q^2) * eta(q^3) * eta(q^12)))^3 in powers of q.
Euler transform of period 12 sequence [-3, 0, 0, -3, -3, -3, -3, -3, 0, 0, -3, -3, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 6^(3/2) (t/i)^(3/2) g(t) where q = exp(2 Pi i t) and g(t) is g.f. for A245669.
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EXAMPLE
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G.f. = 1 - 3*q + 3*q^2 - q^3 - 3*q^4 + 6*q^5 - 3*q^6 + 3*q^8 + 3*q^9 + ...
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MATHEMATICA
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a[ n_] := SeriesCoefficient[ EllipticTheta[3, Pi/3, q]^3, {q, 0, n}];
a[ n_] := SeriesCoefficient[ ((3 EllipticTheta[3, 0, q^9] - EllipticTheta[3, 0, q]) / 2)^3, {q, 0, n}];
a[ n_] := SeriesCoefficient[ (QPochhammer[-q^3, q^6] EllipticTheta[2, 0, Sqrt[-q]] / (2 (-q)^(1/8)))^3, {q, 0, n}];
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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) * eta(x^4 + A) * eta(x^6 + A)^2 / (eta(x^2 + A) * eta(x^3 + A) * eta(x^12 + A)))^3, n))};
(Magma) A := Basis( ModularForms( Gamma0(12), 3/2), 67); A[1] - 3*A[2] + 3*A[3];
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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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