A262364 Expansion of Product_{k>=1} (1-x^(5*k))/(1-x^(2*k)).

1, 0, 1, 0, 2, -1, 3, -1, 5, -2, 6, -3, 10, -5, 13, -7, 19, -11, 25, -15, 35, -22, 45, -30, 62, -41, 79, -55, 105, -75, 134, -98, 175, -130, 220, -168, 284, -219, 355, -280, 451, -360, 561, -455, 705, -578, 870, -725, 1085, -910, 1331, -1132, 1644, -1410

Offset: 0

Author

Vaclav Kotesovec, Sep 23 2015

Keywords

• sign

Comments

In general, if m >= 1 and g.f. = Product_{k>=1} (1-x^((2*m+1)*k))/(1-x^(2*k)), then a(n) ~ (-1)^n * exp(Pi*sqrt((4*m+1)*n/(6*(2*m+1)))) * (4*m+1)^(1/4) / (2^(7/4) * 3^(1/4) * (2*m+1)^(3/4) * n^(3/4)).

Examples

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

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..5000
• 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. A035959, A147783, A262346.

Programs

Mathematica

nmax = 60; CoefficientList[Series[Product[(1-x^(5*k))/(1-x^(2*k)), {k, 1, nmax}], {x, 0, nmax}], x]
CoefficientList[Series[QPochhammer[x^5]/QPochhammer[x^2], {x, 0, 60}], x]

PARI

lista(nn) = {q='q+O('q^nn); Vec(eta(q^5)/eta(q^2))} \\ Altug Alkan, Mar 21 2018

Formula

a(n) ~ (-1)^n * 3^(1/4) * exp(Pi*sqrt(3*n/10)) / (2^(7/4) * 5^(3/4) * n^(3/4)).