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A227587
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Expansion of (phi(-q^3)^2 / (phi(-q) * phi(-q^9)))^2 in powers of q where phi() is a Ramanujan theta function.
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2
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1, 4, 12, 24, 44, 72, 120, 192, 300, 456, 680, 1008, 1464, 2104, 2976, 4176, 5804, 7992, 10920, 14800, 19944, 26688, 35504, 46944, 61752, 80828, 105288, 136536, 176288, 226728, 290448, 370720, 471468, 597600, 755032, 950976, 1194216, 1495352, 1867344, 2325648
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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 1 + 4 * q * chi(-q^3)^2 / (chi(-q) * chi(-q^9))^3 in powers of q where chi() is a Ramanujan theta function.
Expansion of 1 + 4 * b(q^2) * c(q^6) / (b(q) * c(q^3)) in powers of q where b(), c() are cubic AGM theta functions.
Expansion of (eta(q^2) * eta(q^3)^4 * eta(q^18) / (eta(q) * eta(q^6) * eta(q^9))^2)^2 in powers of q.
Euler transform of period 18 sequence [4, 2, -4, 2, 4, -2, 4, 2, 0, 2, 4, -2, 4, 2, -4, 2, 4, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (18 t)) = g(t) where q = exp(2 Pi i t) and g() is the g.f. for A215412.
a(n) ~ exp(2*Pi*sqrt(2*n)/3) / (2^(3/4) * sqrt(3) * n^(3/4)). - Vaclav Kotesovec, Jul 11 2016
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EXAMPLE
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G.f. = 1 + 4*q + 12*q^2 + 24*q^3 + 44*q^4 + 72*q^5 + 120*q^6 + 192*q^7 + 300*q^8 + ...
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MATHEMATICA
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nmax = 50; CoefficientList[Series[Product[((1-x^(2*k)) * (1-x^(3*k))^4 * (1-x^(18*k)) / ((1-x^k) * (1-x^(6*k)) * (1-x^(9*k)))^2)^2, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 11 2016 *)
eta[q_] := q^(1/24)*QPochhammer[q]; CoefficientList[Series[(eta[q^2]* eta[q^3]^4*eta[q^18]/(eta[q]*eta[q^6]*eta[q^9])^2)^2, {q, 0, 50}], q] (* G. C. Greubel, Aug 09 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) * eta(x^3 + A)^4 * eta(x^18 + A) / (eta(x + A) * eta(x^6 + A) * eta(x^9 + A))^2)^2, n))};
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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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