A268500 -id:A268500 - cp's OEIS frontend

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

Vaclav Kotesovec, Feb 06 2016

sign

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

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

Cf. A032302, A264686, A268499, A268500.

Cf. A006951, A100471, A133121.

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

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

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

Original entry on oeis.org

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

Vaclav Kotesovec, Feb 06 2016

sign

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).

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

Cf. A032308, A266821, A268498, A268500.

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

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... .

A268501 Expansion of Product_{k>=1} ((1 + k*x^k) / (1 + x^k))^k.

Original entry on oeis.org

1, 0, 2, 6, 11, 32, 60, 148, 279, 690, 1312, 2778, 5684, 11282, 22920, 44724, 87919, 168978, 329800, 623086, 1189794, 2235744, 4189442, 7795642, 14438670, 26577246, 48616050, 88724110, 160629612, 290267100, 521225220, 933031364, 1661954928, 2950946220

Offset: 0

Vaclav Kotesovec, Feb 06 2016

nonn

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

Cf. A022629, A255528, A266891, A268500, A268502.

nmax = 50; CoefficientList[Series[Product[((1+k*x^k)/(1+x^k))^k, {k, 1, nmax}], {x, 0, nmax}], x]

A268502 Expansion of Product_{k>=1} ((1 + k*x^k) / (1 - x^k))^k.

Original entry on oeis.org

1, 2, 8, 26, 73, 210, 558, 1460, 3663, 9090, 21846, 51690, 120140, 274480, 618656, 1374792, 3017867, 6546610, 14053312, 29852658, 62825894, 131025056, 270948160, 555811298, 1131498850, 2286780266, 4589706604, 9151298134, 18131193484, 35706460678, 69910352496

Offset: 0

Vaclav Kotesovec, Feb 06 2016

nonn

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

Cf. A000219, A022629, A266891, A267004, A268500, A268501.

nmax = 50; CoefficientList[Series[Product[((1+k*x^k)/(1-x^k))^k, {k, 1, nmax}], {x, 0, nmax}], x]

A269339 Expansion of Product_{k>=1} (1 - k*x^k) / (1 + x^k).

Original entry on oeis.org

1, -2, -1, 0, 2, 6, -3, 12, -13, -2, -15, 2, -65, 44, 37, -90, 134, 26, 334, -270, 66, 18, 774, -1280, -15, -2266, 2627, -352, -3575, -516, -484, 5660, -3629, 21408, -20639, -1228, 15595, 31796, -22214, 55390, -104447, 58958, -160254, 180704, 17402, -103200

Offset: 0

Vaclav Kotesovec, Feb 24 2016

sign

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

Cf. A022661, A081362, A267004, A267005, A268500.

nmax=50; CoefficientList[Series[Product[(1-k*x^k)/(1+x^k), {k, 1, nmax}], {x, 0, nmax}], x]

cp's OEIS Frontend

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

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

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A268501 Expansion of Product_{k>=1} ((1 + k*x^k) / (1 + x^k))^k.

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A268502 Expansion of Product_{k>=1} ((1 + k*x^k) / (1 - x^k))^k.

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A269339 Expansion of Product_{k>=1} (1 - k*x^k) / (1 + x^k).

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