A268499 Expansion of Product_{k>=1} ((1 + 3*x^k) / (1 + x^k)).

1, 2, 0, 8, -2, 8, 16, 8, 8, 10, 80, -8, 72, -24, 144, 128, 134, 40, 224, 120, 232, 688, 176, 696, 32, 1194, -96, 1840, 1144, 2248, 288, 2968, 800, 4160, 752, 5104, 6438, 4984, 5104, 5488, 10960, 4856, 14080, 3480, 24408, 15448, 26832, 7080, 42120, 11178

Offset: 0

Author

Vaclav Kotesovec, Feb 06 2016

Keywords

• sign

Comments

In general, for m > 0, if g.f. = Product_{k>=1} ((1 + m*x^k) / (1 + x^k)) then a(n) ~ c^(1/4) * exp(sqrt(c*n)) / (2*sqrt((m+1)*Pi) * n^(3/4)), where c = Pi^2/3 + 2*log(m)^2 + 4*polylog(2, -1/m).

Links

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

Crossrefs

Cf. A032308, A266821, A268498, A268500.

Programs

Mathematica

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

Formula

a(n) ~ c^(1/4) * exp(sqrt(c*n)) / (4*sqrt(Pi)*n^(3/4)), where c = Pi^2/3 + 2*log(3)^2 + 4*polylog(2, -1/3) = 4.467633549370382939364... .