A292560 Expansion of Product_{k>=1} 1/(1 + x^(k^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

Author

Ilya Gutkovskiy, Sep 19 2017

Keywords

• sign

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

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..20000
• Index entries for sequences related to sums of cubes
• Index entries for related partition-counting sequences

Crossrefs

Cf. A081362 (m=1), A292520 (m=2).

Cf. A003108, A279329, A279484.

Programs

Mathematica

nmax = 100; CoefficientList[Series[Product[1/(1 + x^(k^3)), {k, 1, Floor[nmax^(1/3)] + 1}], {x, 0, nmax}], x]

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