A292563 Expansion of Product_{k>=1} (1 + x^((2*k-1)^3)) / (1 - x^((2*k-1)^3)).

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, 4, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 8, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10

Offset: 0

Author

Vaclav Kotesovec, Sep 19 2017

Keywords

• nonn

Comments

Convolution of A292547 and A287091.

In general, if m > 0 and g.f. = Product_{k>=1} (1 + x^((2*k-1)^m)) / (1 - x^((2*k-1)^m)), then a(n) ~ exp((m+1) * ((2^(1 + 1/m)-1) * Gamma(1/m) * Zeta(1 + 1/m)/m^2)^(m/(m+1)) * n^(1/(m+1)) / 2) * ((2^(1 + 1/m)-1) * Gamma(1/m) * Zeta(1 + 1/m))^(m/(2*(m+1))) / (sqrt(Pi*(m+1)) * 2^(m/2 + 1) * m^((m-1)/(2*(m+1))) * n^((2*m+1)/(2*(m+1)))).

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A080054 (m=1), A104274 (m=2).

Cf. A287091, A292547.

Programs

Mathematica

nmax = 100; CoefficientList[Series[Product[(1 + x^((2*k-1)^3)) / (1 - x^((2*k-1)^3)), {k, 1, Floor[nmax^(1/3)/2] + 1}], {x, 0, nmax}], x]

Formula

a(n) ~ exp(2 * ((2^(4/3)-1) * Gamma(1/3) * Zeta(4/3))^(3/4) * n^(1/4) / 3^(3/2)) * ((2^(4/3)-1) * Gamma(1/3) * Zeta(4/3))^(3/8) / (2^(7/2) * 3^(1/4) * sqrt(Pi) * n^(7/8)).