A284628 -id:A284628 - cp's OEIS frontend

A284474 Expansion of Product_{k>=1} (1 + x^(2k))^(2k)/(1 + x^(2k-1))^(2k-1).

1, -1, 3, -6, 11, -22, 42, -74, 131, -231, 395, -669, 1122, -1851, 3029, -4915, 7891, -12572, 19881, -31203, 48657, -75391, 116096, -177792, 270822, -410394, 618905, -929052, 1388403, -2066140, 3062270, -4520912, 6649463, -9745072, 14232278, -20716355, 30057438

Offset: 0

Seiichi Manyama, Apr 15 2017

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Vaclav Kotesovec, Table of n, a(n) for n = 0..2000

Cf. A026011, A281781, A284467, A284628.

nmax = 50; CoefficientList[Series[Product[(1 + x^(2*k))^(2*k)/(1 + x^(2*k-1))^(2*k-1), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Apr 15 2017 *)

a(n) ~ (-1)^n * exp(-1/12 + 3 * 2^(-5/3) * (7*Zeta(3))^(1/3) * n^(2/3)) * A * (7*Zeta(3))^(5/36) / (2^(10/9) * sqrt(3*Pi) * n^(23/36)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Apr 17 2017

G.f.: exp(Sum_{k>=1} (-1)^k*x^k/(k*(1 + x^k)^2)). - Ilya Gutkovskiy, Jun 20 2018

A285287 Expansion of Product_{k>=0} 1/(1 + x^(4k+1))^(4k+1).

Original entry on oeis.org

1, -1, 1, -1, 1, -6, 6, -6, 6, -15, 30, -30, 30, -43, 88, -123, 123, -140, 250, -385, 455, -476, 678, -1098, 1413, -1564, 1913, -2918, 4048, -4707, 5452, -7572, 10747, -13265, 15195, -19534, 27349, -35146, 41042, -50011, 67596, -88897, 106519, -126635, 164230

Offset: 0

Seiichi Manyama, Apr 16 2017

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

Product_{k>=0} 1/(1 + x^(m*k+1))^(m*k+1): A284628 (m=2), A285286 (m=3), this sequence (m=4).

Cf. A285048, A285288.

nmax = 50; CoefficientList[Series[Product[1/(1 + x^(4*k-3))^(4*k-3), {k,1,nmax}], {x,0,nmax}], x] (* Vaclav Kotesovec, Apr 16 2017 *)

a(n) = (-1)^n * A285048(n).

A292038 Expansion of Product_{k>=1} ((1 + x^(2k-1)) / (1 - x^(2k-1)))^(2*k-1).

Original entry on oeis.org

1, 2, 2, 8, 14, 24, 52, 84, 158, 274, 464, 800, 1316, 2208, 3576, 5832, 9358, 14876, 23614, 36936, 57752, 89336, 137716, 210844, 321148, 486890, 733912, 1102336, 1646736, 2451464, 3632832, 5363988, 7889710, 11562712, 16888748, 24581904, 35670242, 51591096

Offset: 0

Vaclav Kotesovec, Sep 08 2017

nonn

Convolution of A262736 and A262811.

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A080054, A262736, A262811, A284628, A285069, A292038.

nmax = 50; CoefficientList[Series[Product[((1+x^(2*k-1))/(1-x^(2*k-1)))^(2*k-1), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ exp(3*(7*Zeta(3))^(1/3)*n^(2/3) / 2^(5/3) - 1/12) * A * (7*Zeta(3))^(5/36) / (2^(31/36) * sqrt(3*Pi) * n^(23/36)), where A is the Glaisher-Kinkelin constant A074962.

G.f.: exp(2*Sum_{k>=1} sigma_2(2*k - 1)*x^(2*k-1)/(2*k - 1)). - Ilya Gutkovskiy, Apr 19 2019

A285311 Expansion of Product_{k>=0} 1/(1 + x^(4k+3))^(4k+3).

Original entry on oeis.org

1, 0, 0, -3, 0, 0, 6, -7, 0, -10, 21, -11, 15, -42, 61, -36, 70, -150, 150, -124, 278, -441, 375, -468, 909, -1131, 1018, -1581, 2602, -2810, 2947, -4819, 6768, -6980, 8509, -13389, 16788, -17609, 23722, -34720, 40337, -44863, 63128, -85430, 95887, -114037

Offset: 0

Seiichi Manyama, Apr 16 2017

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

Product_{k>=0} 1/(1 + x^(m*k+m-1))^(m*k+m-1): A284628 (m=2), A285310 (m=3), this sequence (m=4).

Cf. A285131.

a(n) = (-1)^n * A285131(n).

A285286 Expansion of Product_{k>=0} 1/(1 + x^(3k+1))^(3k+1).

Original entry on oeis.org

1, -1, 1, -1, -3, 3, -3, -4, 14, -14, 4, 24, -44, 31, 37, -107, 126, -4, -208, 329, -175, -319, 777, -704, -236, 1507, -1945, 430, 2532, -4575, 2781, 3236, -9301, 8697, 2085, -16902, 21804, -5233, -26573, 47225, -27047, -34332, 92242, -80162, -26926, 162426

Offset: 0

Seiichi Manyama, Apr 16 2017

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

Product_{k>=0} 1/(1 + x^(m*k+1))^(m*k+1): A284628 (m=2), this sequence (m=3), A285287 (m=4).

Cf. A262949.

nmax = 50; CoefficientList[Series[Product[1/(1 + x^(3*k-2))^(3*k-2), {k,1,nmax}], {x,0,nmax}], x] (* Vaclav Kotesovec, Apr 16 2017 *)

A285294 Expansion of Product_{k>=1} (1 + x^(3k))^(3k) / (1 + x^k)^k.

Original entry on oeis.org

1, -1, -1, 1, -2, -3, 7, -4, -1, 20, -9, -15, 45, -39, -38, 95, -81, -99, 244, -196, -188, 538, -371, -421, 1256, -823, -820, 2575, -1672, -1904, 5367, -3714, -3861, 10555, -7362, -8159, 21391, -14975, -15592, 41654, -28293, -30748, 82026, -54899, -57331, 155933

Offset: 0

Seiichi Manyama, Apr 16 2017

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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), this sequence (m=3), A285295 (m=4).

Cf. A262924.

nmax = 50; CoefficientList[Series[Product[(1 + x^(3*k))^(3*k) / (1 + x^k)^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

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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

A285310 Expansion of Product_{k>=0} 1/(1 + x^(3k+2))^(3k+2).

Original entry on oeis.org

1, 0, -2, 0, 3, -5, -4, 10, -3, -15, 25, 9, -47, 37, 55, -118, 28, 182, -231, -88, 484, -351, -474, 1047, -306, -1479, 1985, 370, -3657, 3120, 2757, -7868, 3686, 8889, -14950, 1255, 22540, -25069, -9557, 49333, -35638, -40172, 96943, -37509, -111145, 172256

Offset: 0

Seiichi Manyama, Apr 16 2017

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

Product_{k>=0} 1/(1 + x^(m*k+m-1))^(m*k+m-1): A284628 (m=2), this sequence (m=3), A285311 (m=4).

cp's OEIS Frontend

A284474 Expansion of Product_{k>=1} (1 + x^(2*k))^(2*k)/(1 + x^(2*k-1))^(2*k-1).

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A285287 Expansion of Product_{k>=0} 1/(1 + x^(4*k+1))^(4*k+1).

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A292038 Expansion of Product_{k>=1} ((1 + x^(2*k-1)) / (1 - x^(2*k-1)))^(2*k-1).

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A285311 Expansion of Product_{k>=0} 1/(1 + x^(4*k+3))^(4*k+3).

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A285286 Expansion of Product_{k>=0} 1/(1 + x^(3*k+1))^(3*k+1).

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A285294 Expansion of Product_{k>=1} (1 + x^(3*k))^(3*k) / (1 + x^k)^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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A285310 Expansion of Product_{k>=0} 1/(1 + x^(3*k+2))^(3*k+2).

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A284474 Expansion of Product_{k>=1} (1 + x^(2k))^(2k)/(1 + x^(2k-1))^(2k-1).

A285287 Expansion of Product_{k>=0} 1/(1 + x^(4k+1))^(4k+1).

A292038 Expansion of Product_{k>=1} ((1 + x^(2k-1)) / (1 - x^(2k-1)))^(2*k-1).

A285311 Expansion of Product_{k>=0} 1/(1 + x^(4k+3))^(4k+3).

A285286 Expansion of Product_{k>=0} 1/(1 + x^(3k+1))^(3k+1).

A285294 Expansion of Product_{k>=1} (1 + x^(3k))^(3k) / (1 + x^k)^k.

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

A285310 Expansion of Product_{k>=0} 1/(1 + x^(3k+2))^(3k+2).