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A259884
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Expansion of phi(x) * f(-x^3)^3 / f(-x) in powers of x where phi(), f() are Ramanujan theta functions.
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1
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1, 3, 4, 4, 4, 7, 8, 4, 5, 8, 12, 8, 4, 12, 8, 8, 9, 12, 16, 4, 12, 15, 8, 8, 8, 20, 20, 8, 8, 12, 16, 16, 8, 15, 20, 12, 12, 16, 16, 12, 13, 24, 20, 8, 8, 24, 24, 8, 16, 12, 28, 16, 12, 28, 8, 20, 13, 20, 28, 16, 20, 24, 16, 8, 8, 27, 36, 12, 16, 28, 24, 12
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OFFSET
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0,2
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LINKS
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FORMULA
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Expansion of phi(x) * c(x) * (1/3) * x^(-1/3) in powers of x where phi() is a Ramanujan theta function (A000122) and c() is a cubic AGM theta function (A005882).
Expansion of q^(-1/3) * eta(q^2)^5 * eta(q^3)^3 / (eta(q)^3 * eta(q^4)^2) in powers of q.
Euler transform of period 12 sequence [3, -2, 0, 0, 3, -5, 3, 0, 0, -2, 3, -3, ...].
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EXAMPLE
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G.f. = 1 + 3*x + 4*x^2 + 4*x^3 + 4*x^4 + 7*x^5 + 8*x^6 + 4*x^7 + 5*x^8 + ...
G.f. = q + 3*q^4 + 4*q^7 + 4*q^10 + 4*q^13 + 7*q^16 + 8*q^19 + 4*q^22 + ...
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MATHEMATICA
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eta[q_]:= q^(1/24)*QPochhammer[q]; a[n_] := SeriesCoefficient[q^(-1/3)* eta[q^2]^5*eta[q^3]^3/(eta[q]^3*eta[q^4]^2), {q, 0, n}]; Table[a[n], {n, 0, 50}] (* G. C. Greubel, Mar 16 2018 *)
a[n_]:= SeriesCoefficient[EllipticTheta[3, 0, q]*QPochhammer[q^3]^3 /QPochhammer[q], {q, 0, n}]; Table[a[n], {n, 0, 50}] (* G. C. Greubel, Mar 18 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^2 + A)^5 * eta(x^3 + A)^3 / (eta(x + A)^3 * eta(x^4 + A)^2), n))};
(PARI) q='q+O('q^99); Vec(eta(q^2)^5*eta(q^3)^3/(eta(q)^3*eta(q^4)^2)) \\ Altug Alkan, Mar 18 2018
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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