A268498 Expansion of Product_{k>=1} ((1 + 2*x^k) / (1 + x^k)).

1, 1, 0, 3, -1, 3, 3, 3, 0, 4, 12, 0, 9, -3, 21, 12, 17, -3, 33, 0, 33, 36, 36, 27, 21, 52, 24, 90, 72, 99, 24, 138, 21, 207, 0, 261, 149, 267, 45, 333, 174, 339, 174, 345, 411, 654, 330, 456, 657, 535, 684, 483, 1233, 489, 1353, 882, 1803, 720, 1902, 756

Offset: 0

Author

Vaclav Kotesovec, Feb 06 2016

Keywords

• sign

Comments

It appears that this sequence contains only finitely many nonpositive terms, namely at indices {2, 4, 8, 11, 13, 17, 19, 34}. - Gus Wiseman, Jan 23 2019

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..5000
• Vaclav Kotesovec, Graph - The asymptotic ratio

Crossrefs

Cf. A032302, A264686, A268499, A268500.

Cf. A006951, A100471, A133121.

Programs

Mathematica

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

Formula

a(n) ~ c^(1/4) * exp(sqrt(c*n)) / (2*sqrt(3*Pi)*n^(3/4)), where c = Pi^2/3 + 2*log(2)^2 + 4*polylog(2, -1/2) = 2.4571173338382709125... .

a(n) = Sum_{k = 0...n} (-1)^k * A133121(n,k). - Gus Wiseman, Jan 23 2019

G.f.: Product_{k>=1} (1 - Sum_{j>=1} (-1)^j * x^(k*j)). - Ilya Gutkovskiy, Nov 06 2019