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A139632
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Expansion of chi(q) * chi(-q^5) in powers of q where chi() is a Ramanujan theta function.
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5
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1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 2, 3, 3, 2, 3, 4, 3, 2, 4, 5, 4, 4, 5, 6, 6, 5, 6, 8, 7, 6, 8, 11, 10, 8, 11, 13, 11, 10, 13, 16, 15, 14, 17, 20, 18, 17, 20, 24, 23, 21, 25, 31, 29, 26, 32, 37, 34, 32, 39, 44, 42, 41, 47, 54, 52, 49
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OFFSET
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0,13
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COMMENTS
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LINKS
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FORMULA
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Expansion of q^(1/4) * eta(q^2)^2 * eta(q^5) / (eta(q) * eta(q^4) * eta(q^10)) in powers of q.
Euler transform of period 20 sequence [ 1, -1, 1, 0, 0, -1, 1, 0, 1, -1, 1, 0, 1, -1, 0, 0, 1, -1, 1, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (640 t)) = 2^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A139631.
G.f.: Product_{k>0} (1 + x^k) / ((1 + x^(2*k)) * (1 + x^(5*k))).
a(n) ~ exp(Pi*sqrt(n/10)) / (2^(5/4) * 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^3 + x^4 + x^7 + x^8 + x^9 + x^10 + x^11 + 2*x^12 + x^13 + ...
G.f. = 1/q + q^3 + q^11 + q^15 + q^27 + q^31 + q^35 + q^39 + q^43 + 2*q^47 + ...
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MATHEMATICA
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a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x^2] QPochhammer[ x^5, x^10], {x, 0, n}]; (* Michael Somos, Sep 07 2015 *)
nmax = 100; CoefficientList[Series[Product[(1 + x^k) / ((1 + x^(2*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^2 + A)^2 * eta(x^5 + A) / (eta(x + A) * eta(x^4 + A) * eta(x^10 + 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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