A262924 -id:A262924 - 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)).

A262923 Expansion of Product_{k>=1} 1 / ((1-x^(3k-1))^(3k-1) * (1-x^(3k-2))^(3k-2)).

Original entry on oeis.org

1, 1, 3, 3, 10, 15, 27, 44, 79, 128, 211, 331, 549, 843, 1338, 2061, 3195, 4851, 7384, 11104, 16696, 24774, 36817, 54173, 79560, 116067, 168880, 244293, 352480, 506012, 724531, 1032762, 1468271, 2079525, 2937102, 4134399, 5804795, 8124459, 11342952, 15791650

Offset: 0

Vaclav Kotesovec, Oct 04 2015

nonn

Convolution of A262946 and A262947.

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. A262883, A262876, A262877, A262924, A035386, A035382, A262946, A262947.

nmax=60; CoefficientList[Series[Product[1/((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(-1/6 + 3^(2/3)*(Zeta(3)/2)^(1/3) * n^(2/3)) * A^2 * Zeta(3)^(1/9) / (2^(5/18) * 3^(31/36) * sqrt(Pi) * n^(11/18)), where Zeta(3) = A002117 and A = A074962 is the Glaisher-Kinkelin constant.

A262884 Expansion of Product_{k>=1} ((1+x^(3k-1))(1+x^(3*k-2)))^k.

Original entry on oeis.org

1, 1, 1, 1, 2, 4, 4, 7, 9, 11, 16, 23, 31, 40, 53, 71, 91, 121, 161, 206, 264, 343, 441, 563, 725, 922, 1166, 1476, 1869, 2357, 2967, 3725, 4659, 5816, 7263, 9050, 11241, 13947, 17269, 21333, 26342, 32479, 39957, 49094, 60231, 73775, 90273, 110333, 134643

Offset: 0

Vaclav Kotesovec, Oct 04 2015

nonn

Convolution of A262878 and A262879.

Vaclav Kotesovec, Table of n, a(n) for n = 0..2000
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. A262878, A262879, A262883, A262924.

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

a(n) ~ exp(-Pi^4/(2592*Zeta(3)) + Pi^2 * n^(1/3) / (12*3^(2/3)*Zeta(3)^(1/3)) + 3^(2/3) * Zeta(3)^(1/3) * n^(2/3)/2) * Zeta(3)^(1/6) / (2^(7/18) * 3^(2/3) * sqrt(Pi) * n^(2/3)).

A262949 Expansion of Product_{k>=1} (1 + x^(3k-2))^(3k-2).

Original entry on oeis.org

1, 1, 0, 0, 4, 4, 0, 7, 13, 6, 10, 38, 32, 17, 74, 103, 59, 139, 266, 191, 247, 593, 581, 513, 1175, 1487, 1190, 2223, 3453, 2938, 4158, 7264, 7095, 8052, 14430, 16308, 16246, 27364, 35347, 34096, 50997, 72595, 72163, 94707, 142522, 151435, 178047, 270112

Offset: 0

Vaclav Kotesovec, Oct 05 2015

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A261612, A262879, A262884, A262924, A262948.

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

a(n) ~ exp(3 * 2^(-4/3) * Zeta(3)^(1/3) * n^(2/3)) * Zeta(3)^(1/6) / (2^(7/12) * sqrt(3*Pi) * n^(2/3)).

A262948 Expansion of Product_{k>=1} (1 + x^(3k-1))^(3k-1).

Original entry on oeis.org

1, 0, 2, 0, 1, 5, 0, 10, 8, 5, 26, 11, 28, 62, 24, 101, 111, 77, 260, 202, 268, 583, 382, 761, 1165, 847, 1940, 2198, 2061, 4346, 4084, 5078, 9039, 7844, 11978, 17620, 15721, 26648, 33219, 32894, 56000, 61494, 69653, 111884, 114265, 146557, 214864, 214967

Offset: 0

Vaclav Kotesovec, Oct 05 2015

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A262736, A262878, A262884, A262924, A262928, A262949.

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

a(n) ~ exp(3 * 2^(-4/3) * Zeta(3)^(1/3) * n^(2/3)) * Zeta(3)^(1/6) / (2^(7/12) * sqrt(3*Pi) * n^(2/3)).

A285292 Expansion of Product_{k>=1} (1 + x^k)^k / (1 + x^(4k))^(4k).

Original entry on oeis.org

1, 1, 2, 5, 4, 12, 20, 29, 53, 80, 127, 199, 311, 468, 715, 1079, 1621, 2402, 3541, 5210, 7574, 11046, 15926, 22917, 32804, 46766, 66419, 93936, 132331, 185830, 260144, 362752, 504573, 699376, 966842, 1332721, 1832217, 2512209, 3435932, 4687884, 6380911

Offset: 0

Vaclav Kotesovec, Apr 16 2017

nonn

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

Cf. A262736, A262924, A285293.

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

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

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

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), 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 *)

cp's OEIS Frontend

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

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A262923 Expansion of Product_{k>=1} 1 / ((1-x^(3*k-1))^(3*k-1) * (1-x^(3*k-2))^(3*k-2)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A262884 Expansion of Product_{k>=1} ((1+x^(3*k-1))*(1+x^(3*k-2)))^k.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A262949 Expansion of Product_{k>=1} (1 + x^(3*k-2))^(3*k-2).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A262948 Expansion of Product_{k>=1} (1 + x^(3*k-1))^(3*k-1).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A285292 Expansion of Product_{k>=1} (1 + x^k)^k / (1 + x^(4*k))^(4*k).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

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

A262923 Expansion of Product_{k>=1} 1 / ((1-x^(3k-1))^(3k-1) * (1-x^(3k-2))^(3k-2)).

A262884 Expansion of Product_{k>=1} ((1+x^(3k-1))(1+x^(3*k-2)))^k.

A262949 Expansion of Product_{k>=1} (1 + x^(3k-2))^(3k-2).

A262948 Expansion of Product_{k>=1} (1 + x^(3k-1))^(3k-1).

A285292 Expansion of Product_{k>=1} (1 + x^k)^k / (1 + x^(4k))^(4k).

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

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