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A292560
Expansion of Product_{k>=1} 1/(1 + x^(k^3)).
3
1, -1, 1, -1, 1, -1, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, 1, -1, 1, -1, 1, -1, 0, 0, 0, -1, 1, -1, 1, -1, 2, -2, 2, -1, 1, -1, 1, -1, 0, 0, 0, -1, 1, -1, 1, -1, 2, -2, 2, -1, 1, -1, 2, -2, 1, -1, 1, -2, 2, -2, 1, -1, 1, -1, 1, 0, 0, 0, 1, -1, 1, -1, 1, -2, 2, -2, 1, -1, 1, -2, 2, -1, 1, -1, 2, -2, 2, -1, 1
OFFSET
0,33
COMMENTS
Convolution inverse of A279329.
The difference between the number of partitions of n into an even number of cubes and the number of partitions of n into an odd number of cubes.
In general, if m > 0 and g.f. = Product_{k>=1} 1/(1 + x^(k^m)), then a(n) ~ (-1)^n * exp((m+1) * (Gamma(1/m) * Zeta(1 + 1/m) / m^2)^(m/(m+1)) * n^(1/(m+1)) / 2) * (Gamma(1/m) * Zeta(1 + 1/m))^(m/(2*(m+1))) / (sqrt(Pi*(m+1)) * 2^((m+1)/2) * m^((m-1)/(2*(m+1))) * n^((2*m+1)/(2*(m+1)))). - Vaclav Kotesovec, Sep 19 2017
FORMULA
G.f.: Product_{k>=1} 1/(1 + x^(k^3)).
a(n) ~ (-1)^n * exp(2 * (Gamma(1/3) * Zeta(4/3))^(3/4) * n^(1/4) / 3^(3/2)) * (Gamma(1/3) * Zeta(4/3))^(3/8) / (8 * 3^(1/4) * sqrt(Pi) * n^(7/8)). - Vaclav Kotesovec, Sep 19 2017
MATHEMATICA
nmax = 100; CoefficientList[Series[Product[1/(1 + x^(k^3)), {k, 1, Floor[nmax^(1/3)] + 1}], {x, 0, nmax}], x]
CROSSREFS
Cf. A081362 (m=1), A292520 (m=2).
Sequence in context: A372924 A004571 A204429 * A086137 A085976 A171624
KEYWORD
sign
AUTHOR
Ilya Gutkovskiy, Sep 19 2017
STATUS
approved