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A258108
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Expansion of b(-q) * b(q^6) / (b(q^3) * b(q^12)) in powers of q where b() is a cubic AGM theta function.
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3
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1, 3, 0, -3, 6, 0, -12, 15, 0, -30, 36, 0, -60, 78, 0, -117, 150, 0, -228, 276, 0, -420, 504, 0, -732, 885, 0, -1245, 1488, 0, -2088, 2454, 0, -3420, 3996, 0, -5460, 6378, 0, -8583, 9972, 0, -13344, 15378, 0, -20448, 23472, 0, -30876, 35379, 0, -46116, 52644
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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)^9 * eta(q^9) * eta(q^36) / (eta(q)^3 * eta(q^3)^2 * eta(q^4)^3 * eta(q^12)^2 * eta(q^18)) in powers of q.
Euler transform of period 36 sequence [ 3, -6, 5, -3, 3, -4, 3, -3, 4, -6, 3, 1, 3, -6, 5, -3, 3, -4, 3, -3, 5, -6, 3, 1, 3, -6, 4, -3, 3, -4, 3, -3, 5, -6, 3, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 3 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A164616.
a(3*n + 2) = 0. a(3*n + 1) = 3 * A132977(n). a(3*n) = -3 * A164617(n) unless n = 0.
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EXAMPLE
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G.f. = 1 + 3*q - 3*q^3 + 6*q^4 - 12*q^6 + 15*q^7 - 30*q^9 + 36*q^10 + ...
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MATHEMATICA
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QP=QPochhammer; A258108[n_] :=SeriesCoefficient[(QP[x^2]^9*QP[x^9]*QP[x^36])/(QP[x]^3*QP[x^3]^2*QP[x^4]^3*QP[x^12]^2*QP[x^18]), {x, 0, n}]; Table[A258108[n], {n, 0, 50}] (* G. C. Greubel, Oct 18 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^2 + A)^9 * eta(x^9 + A) * eta(x^36 + A) / (eta(x + A)^3 * eta(x^3 + A)^2 * eta(x^4 + A)^3 * eta(x^12 + A)^2 * eta(x^18 + A)), 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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