A285292 -id:A285292 - cp's OEIS frontend

A262736 Expansion of Product_{k>=1} (1 + x^(2k-1))^(2k-1).

1, 1, 0, 3, 3, 5, 8, 10, 22, 25, 41, 57, 88, 126, 168, 261, 351, 512, 685, 984, 1357, 1865, 2566, 3485, 4838, 6459, 8832, 11831, 16056, 21404, 28660, 38259, 50875, 67613, 89161, 118184, 155321, 204609, 267708, 351125, 458331, 597740, 777590, 1010020, 1310390

Offset: 0

Vaclav Kotesovec, Sep 29 2015

nonn

Alois P. Heinz, 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], Sep 30 2015, p. 20.

Cf. A026007, A026011, A255528, A262811, A292038.

Cf. A262924, A285292, A285293.

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

a(n) ~ exp(3^(4/3) * (Zeta(3))^(1/3) * n^(2/3) / 2^(5/3)) * Zeta(3)^(1/6) / (2^(3/4) * 3^(1/3) * sqrt(Pi) * n^(2/3)).

A262924 Expansion of Product_{k>=1} (1 + x^(3k-1))^(3k-1) * (1 + x^(3k-2))^(3k-2).

Original entry on oeis.org

1, 1, 2, 2, 5, 10, 13, 25, 35, 57, 87, 134, 211, 306, 458, 684, 996, 1465, 2129, 3073, 4411, 6288, 8977, 12707, 17913, 25185, 35231, 49078, 68228, 94490, 130408, 179425, 246121, 336681, 459239, 624842, 847986, 1147728, 1549773, 2087972, 2806455, 3764136

Offset: 0

Vaclav Kotesovec, Oct 04 2015

nonn

Convolution of A262948 and A262949.

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Vaclav Kotesovec)
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015

Cf. A262884, A262878, A262879, A262923, A261612, A262928, A262948, A262949.

Cf. A262736, A285292, A285293.

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

a(n) ~ exp(3*Zeta(3)^(1/3)*n^(2/3)/2) * Zeta(3)^(1/6) / (2^(1/3) * sqrt(3*Pi) * n^(2/3)).

A285293 Expansion of Product_{k>=1} (1 + x^k)^k / (1 + x^(5k))^(5k).

Original entry on oeis.org

1, 1, 2, 5, 8, 11, 23, 39, 58, 102, 160, 250, 392, 614, 929, 1426, 2155, 3221, 4816, 7124, 10516, 15389, 22448, 32549, 47027, 67586, 96779, 138052, 196078, 277606, 391570, 550516, 771442, 1077818, 1501214, 2084899, 2887759, 3988792, 5495381, 7552127, 10353345

Offset: 0

Vaclav Kotesovec, Apr 16 2017

nonn

In general, if m > 1 and g.f. = Product_{k>=1} (1 + x^k)^k / (1 + x^(m*k))^(m*k), then a(n, m) ~ exp(2^(-4/3) * 3^(4/3) * (1-1/m)^(1/3) * Zeta(3)^(1/3) * n^(2/3)) * 2^(m/12 - 3/4) * (1-1/m)^(1/6) * Zeta(3)^(1/6) / (3^(1/3) * sqrt(Pi) * n^(2/3)).

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Vaclav Kotesovec)

Cf. A262736 (m=2), A262924 (m=3), A285292 (m=4).

nmax = 50; CoefficientList[Series[Product[(1+x^k)^k/(1+x^(5*k))^(5*k), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ exp(2^(-2/3) * 3^(4/3) * 5^(-1/3) * Zeta(3)^(1/3) * n^(2/3)) * Zeta(3)^(1/6) / (3^(1/3) * 5^(1/6) * sqrt(Pi) * n^(2/3)).

A285284 Expansion of Product_{k>=1} (1 - x^k)^k/(1 - x^(4k))^(4k).

Original entry on oeis.org

1, -1, -2, -1, 4, 0, -4, 3, 21, -4, -29, -7, 51, -24, -105, -7, 201, -30, -291, 34, 642, -42, -874, 75, 1764, -262, -2737, -40, 4555, -818, -7512, 88, 12425, -1492, -19062, 1135, 32637, -2573, -47688, 3576, 81335, -6477, -119540, 6525, 193738, -18478, -292685

Offset: 0

Seiichi Manyama, Apr 16 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Product_{k>=1} (1 - x^k)^k/(1 - x^(m*k))^(m*k): A285069 (m=2), A285247 (m=3), this sequence (m=4), A285285 (m=5).

Cf. A285215, A285292.

nmax = 50; CoefficientList[Series[Product[(1 - x^k)^k/(1 - x^(4*k))^(4*k), {k,1,nmax}], {x,0,nmax}], x] (* Vaclav Kotesovec, Apr 16 2017 *)

A285295 Expansion of Product_{k>=1} (1 + x^(4k))^(4k) / (1 + x^k)^k.

Original entry on oeis.org

1, -1, -1, -2, 5, -4, 0, -6, 26, -16, 6, -31, 93, -81, 19, -147, 310, -295, 136, -486, 1069, -940, 645, -1575, 3338, -3021, 2301, -5089, 9735, -9381, 7548, -15506, 27556, -27587, 23664, -44862, 76043, -77620, 70982, -124744, 204389, -211376, 203644, -336775

Offset: 0

Seiichi Manyama, Apr 16 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Product_{k>=1} (1 + x^(m*k))^(m*k) / (1 + x^k)^k: A284628 (m=2), A285294 (m=3), this sequence (m=4).

Cf. A285292.

nmax = 50; CoefficientList[Series[Product[(1 + x^(4*k))^(4*k) / (1 + x^k)^k, {k,1,nmax}], {x,0,nmax}], x] (* Vaclav Kotesovec, Apr 16 2017 *)

a(n) ~ (-1)^n * exp(-1/12 + 3 * (5*Zeta(3))^(1/3) * n^(2/3) / 4) * A * (5*Zeta(3))^(5/36) / (2^(5/4) * sqrt(3*Pi) * n^(23/36)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Apr 17 2017

cp's OEIS Frontend

A262736 Expansion of Product_{k>=1} (1 + x^(2*k-1))^(2*k-1).

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A262924 Expansion of Product_{k>=1} (1 + x^(3*k-1))^(3*k-1) * (1 + x^(3*k-2))^(3*k-2).

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A285293 Expansion of Product_{k>=1} (1 + x^k)^k / (1 + x^(5*k))^(5*k).

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A285284 Expansion of Product_{k>=1} (1 - x^k)^k/(1 - x^(4*k))^(4*k).

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A285295 Expansion of Product_{k>=1} (1 + x^(4*k))^(4*k) / (1 + x^k)^k.

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Author

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Mathematica

Formula

A262736 Expansion of Product_{k>=1} (1 + x^(2k-1))^(2k-1).

A262924 Expansion of Product_{k>=1} (1 + x^(3k-1))^(3k-1) * (1 + x^(3k-2))^(3k-2).

A285293 Expansion of Product_{k>=1} (1 + x^k)^k / (1 + x^(5k))^(5k).

A285284 Expansion of Product_{k>=1} (1 - x^k)^k/(1 - x^(4k))^(4k).

A285295 Expansion of Product_{k>=1} (1 + x^(4k))^(4k) / (1 + x^k)^k.