A287091 Expansion of Product_{k>=1} 1/(1 - x^((2*k-1)^3)).

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

Offset: 0

Author

Ilya Gutkovskiy, May 19 2017

Keywords

• nonn

Comments

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

Examples

			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].
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
• Index entries for sequences related to sums of cubes
• Index entries for sequences related to partitions

Crossrefs

Cf. A000009 (m=1), A167661 (m=2).

Cf. A003108, A016755, A259792, A279329, A280865.

Cf. A001156, A046042, A292547.

Programs

Mathematica

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 *)

Formula

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