A262346 Expansion of Product_{k>=1} (1-x^(3*k))/(1-x^(2*k)).

1, 0, 1, -1, 2, -1, 2, -2, 4, -3, 5, -5, 8, -7, 10, -10, 15, -14, 19, -20, 27, -26, 34, -36, 47, -47, 59, -63, 79, -81, 99, -106, 130, -135, 162, -174, 208, -219, 258, -278, 328, -347, 404, -436, 507, -540, 621, -671, 772, -825, 941, -1017, 1159, -1242, 1405

Offset: 0

Author

Vaclav Kotesovec, Sep 23 2015

Keywords

• sign

Examples

			G.f. = 1 + x^2 - x^3 + 2*x^4 - x^5 + 2*x^6 - 2*x^7 + 4*x^8 - 3*x^9 + 5*x^10 + ...
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
• Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], 2015-2016, p. 15.

Crossrefs

Cf. A000726, A007690, A262151, A262364.

Programs

Mathematica

nmax = 60; CoefficientList[Series[Product[(1-x^(3*k))/(1-x^(2*k)), {k, 1, nmax}], {x, 0, nmax}], x]
CoefficientList[Series[QPochhammer[x^3]/QPochhammer[x^2], {x, 0, 60}], x]

PARI

lista(nn) = {q='q+O('q^nn); Vec(eta(q^3)/eta(q^2))} \\ Altug Alkan, Mar 21 2018

Formula

a(n) ~ (-1)^n * 5^(1/4) * exp(Pi*sqrt(5*n/2)/3) / (3 * 2^(7/4) * n^(3/4)).