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A128762
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Expansion of chi(q) * chi(q^2) / (chi(q^5) * chi(q^10)) in powers of q where chi() is a Ramanujan theta function.
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1
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1, 1, 1, 2, 1, 1, 2, 2, 2, 4, 4, 4, 5, 5, 6, 6, 8, 9, 10, 12, 14, 15, 17, 20, 21, 23, 26, 30, 32, 37, 42, 44, 50, 56, 60, 66, 74, 80, 88, 98, 109, 119, 130, 144, 154, 167, 184, 200, 218, 241, 262, 284, 308, 334, 362, 390, 426, 462, 498, 542, 589, 633, 685, 742, 796, 858
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OFFSET
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0,4
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COMMENTS
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LINKS
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FORMULA
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Euler transform of period 40 sequence [ 1, 0, 1, -1, 0, 0, 1, 0, 1, 0, 1, -1, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, -1, 1, 0, 1, 0, 1, 0, 0, -1, 1, 0, 1, 0, ...].
Given g.f. A(x), then B(q) = q*A(q^2) satisfies 0 = f(B(q), B(q^3)) where f(u, v) = (u - v^3) * (u^3 - v) - 3*u*v * (u^2 + v^2).
G.f.: Product_{k>0} (1 + x^k) * (1 + x^(20*k)) / ( (1 + x^(4*k)) * (1+x^(5*k))).
a(n) ~ exp(Pi*sqrt(n/5)) / (2^(3/2) * 5^(1/4) * n^(3/4)). - Vaclav Kotesovec, Sep 08 2015
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EXAMPLE
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G.f. = 1 + x + x^2 + 2*x^3 + x^4 + x^5 + 2*x^6 + 2*x^7 + 2*x^8 + 4*x^9 + ...
G.f. = q + q^3 + q^5 + 2*q^7 + q^9 + q^11 + 2*q^13 + 2*q^15 + 2*q^17 + ...
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MATHEMATICA
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a[ n_] := SeriesCoefficient[ (QPochhammer[ x^5, - x^5] QPochhammer[ x^10, -x^10]) / (QPochhammer[ x, -x] QPochhammer[ x^2, -x^2]), {x, 0, n}]; (* Michael Somos, Apr 26 2015 *)
nmax = 40; CoefficientList[Series[Product[(1 + x^k) * (1 + x^(20*k)) / ( (1 + x^(4*k)) * (1+x^(5*k))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 08 2015 *)
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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^8 + A) * eta(x^10 + A) * eta(x^20 + A) / (eta(x^2 + A) * eta(x^4 + A) * eta(x^5 + A) * eta(x^40 + A)), 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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