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A287091
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Expansion of Product_{k>=1} 1/(1 - x^((2*k-1)^3)).
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4
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1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5
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OFFSET
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0,28
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COMMENTS
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Number of partitions of n into odd cubes.
In general, if m > 0 and g.f. = Product_{k>=1} 1/(1 - x^((2*k-1)^m)), then a(n) ~ exp((m+1) * (Gamma(1/m) * Zeta(1+1/m) / (2*m^2))^(m/(m+1)) * n^(1/(m+1))) * (Gamma(1/m) * Zeta(1+1/m))^(m/(2*(m+1))) / (sqrt(Pi*(m+1)) * 2^((1+m*(m+3))/(2*(m+1))) * m^((m-1)/(2*(m+1))) * n^((2*m+1)/(2*(m+1)))). - Vaclav Kotesovec, Sep 19 2017
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LINKS
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FORMULA
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G.f.: Product_{k>=1} 1/(1 - x^((2*k-1)^3)).
a(n) ~ exp(2^(5/4) * (Gamma(1/3) * Zeta(4/3))^(3/4) * n^(1/4) / 3^(3/2)) * (Gamma(1/3) * Zeta(4/3)/2)^(3/8) / (8 * 3^(1/4) * sqrt(Pi) * n^(7/8)). - Vaclav Kotesovec, Sep 18 2017
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EXAMPLE
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a(27) = 2 because we have [27] and [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1].
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MATHEMATICA
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nmax = 110; CoefficientList[Series[Product[1/(1 - x^((2*k-1)^3)), {k, 1, Floor[nmax^(1/3)/2] + 1}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 18 2017 *)
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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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