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A122407
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Expansion of q^(-2/3) * b(q) * b(q^2) * c(q^2) / 3 in powers of q where b(), c() are cubic AGM theta functions.
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4
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1, -3, -2, 12, -4, -9, 6, -12, 8, 3, 4, 12, -16, 33, -24, -60, 7, 27, 8, 24, 44, -24, 18, -36, -34, -48, -12, 84, -40, 9, 24, 48, -33, 21, -16, 0, 72, -111, -6, 36, 50, -96, -8, -120, 8, 111, -24, 96, -16, 195, 32, -132, -76, 48, -66, 24, -54, -123, 48, -144, -32, 102, 120, -24, 176, 117, -14, -12, -28, -192, -54
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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^(-2/3) * eta(q)^3 * eta(q^2)^2 * eta(q^6)^2 / eta(q^3) in powers of q.
Euler transform of period 6 sequence [-3, -5, -2, -5, -3, -6, ...].
Expansion of a newform level 18 weight 3 and character chi_18(17,.). - Michael Somos, Jan 08 2017
a(n) = sqrt(-2) * b(3*n + 2) where b() is multiplicative with b(p^e) = b(p) * b(p^(e-1)) - Kronecker(-12, p) * p^2 * b(p^(e-2)). - Michael Somos, Jan 08 2017
G.f. is a period 1 Fourier series which satisfies f(-1 / (18 t)) = 972^(1/2) (t/i)^3 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A208385.
G.f.: Product_{k>0} (1 - x^k)^6 * (1 + x^k)^4 * (1 - x^k + x^(2*k))^2 *(1 + x^k + x^(2*k)).
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EXAMPLE
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G.f. = 1 - 3*x - 2*x^2 + 12*x^3 - 4*x^4 - 9*x^5 + 6*x^6 - 12*x^7 + 8*x^8 + ...
G.f. = q^2 - 3*q^5 - 2*q^8 + 12*q^11 - 4*q^14 - 9*q^17 + 6*q^20 - 12*q^23 + ...
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MATHEMATICA
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a[ n_] := SeriesCoefficient[ QPochhammer[ x]^3 QPochhammer[ x^2]^2 QPochhammer[ x^6]^2 / QPochhammer[ x^3], {x, 0, n}]; (* Michael Somos, Jan 08 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)^3 * eta(x^2 + A)^2 * eta(x^6 + A)^2 / eta(x^3 + A), n))};
(Magma) A := Basis( CuspForms( Gamma1(18), 3), 211); A[2] - 3*A[5] - 2*A[8]; /* Michael Somos, Jan 08 2017 */
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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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