A262736 -id:A262736 - cp's OEIS frontend

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

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)).

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

A285338 Expansion of Product_{k>=1} (1 + x^(5k-4))^(5k-4).

Original entry on oeis.org

1, 1, 0, 0, 0, 0, 6, 6, 0, 0, 0, 11, 26, 15, 0, 0, 16, 82, 86, 20, 0, 21, 172, 316, 180, 15, 26, 328, 872, 790, 226, 37, 538, 2043, 2681, 1310, 202, 845, 4184, 7426, 5390, 1447, 1290, 7855, 18067, 17705, 7277, 2662, 13723, 39468, 50030, 28707, 8742, 22979, 79760

Offset: 0

Vaclav Kotesovec, Apr 17 2017

nonn

For all n<=30 a(n) = abs(A285071(n)), but a(31) <> abs(A285071(31)).

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

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

Product_{k>=0} (1 + x^(m*k+1))^(m*k+1): A026007 (m=1), A262736 (m=2), A262949 (m=3), A285288 (m=4), this sequence (m=5).

Cf. A285071, A285340.

nmax = 50; CoefficientList[Series[Product[(1+x^(5*k-4))^(5*k-4), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ exp(2^(-4/3) * 3^(4/3) * 5^(-1/3) * Zeta(3)^(1/3) * n^(2/3)) * Zeta(3)^(1/6) / (2^(41/60) * 3^(1/3) * 5^(1/6) * sqrt(Pi) * n^(2/3)).

A285340 Expansion of Product_{k>=0} (1 + x^(5k+4))^(5k+4).

Original entry on oeis.org

1, 0, 0, 0, 4, 0, 0, 0, 6, 9, 0, 0, 4, 36, 14, 0, 1, 54, 92, 19, 0, 36, 228, 202, 24, 9, 272, 702, 358, 29, 158, 1168, 1696, 598, 70, 1027, 3810, 3605, 904, 501, 4600, 10196, 6898, 1408, 3078, 15805, 24104, 12242, 2838, 14103, 46090, 51376, 20566, 9443, 51682

Offset: 0

Seiichi Manyama, Apr 17 2017

nonn

In general, if m > 1 and g.f. = Product_{k>=1} (1 + x^(m*k-1))^(m*k-1), then a(n, m) ~ exp(2^(-4/3) * 3^(4/3) * m^(-1/3) * Zeta(3)^(1/3) * n^(2/3)) * Zeta(3)^(1/6) / (2^(1/6 + 1/(2*m) + m/12) * 3^(1/3) * m^(1/6) * sqrt(Pi) * n^(2/3)). - Vaclav Kotesovec, Apr 17 2017

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

Product_{k>=0} (1 + x^(m*k+m-1))^(m*k+m-1): A262736 (m=2), A262948 (m=3), A285339 (m=4), this sequence (m=5).

Cf. A285214, A285338.

nmax = 50; CoefficientList[Series[Product[(1 + x^(5*k-1))^(5*k-1), {k,1,nmax}], {x,0,nmax}], x] (* Vaclav Kotesovec, Apr 17 2017 *)

a(n) ~ exp(2^(-4/3) * 3^(4/3) * 5^(-1/3) * Zeta(3)^(1/3) * n^(2/3)) * Zeta(3)^(1/6) / (2^(41/60) * 3^(1/3) * 5^(1/6) * sqrt(Pi) * n^(2/3)). - Vaclav Kotesovec, Apr 17 2017

A285339 Expansion of Product_{k>=0} (1 + x^(4k+3))^(4k+3).

Original entry on oeis.org

1, 0, 0, 3, 0, 0, 3, 7, 0, 1, 21, 11, 0, 21, 54, 15, 7, 96, 122, 19, 74, 311, 217, 44, 351, 768, 367, 209, 1227, 1663, 591, 989, 3402, 3225, 1156, 3609, 8289, 5815, 3053, 11096, 18015, 10176, 9466, 29593, 36249, 18454, 28960, 71093, 68438, 37297, 81606

Offset: 0

Seiichi Manyama, Apr 17 2017

nonn

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

Product_{k>=0} (1 + x^(m*k+m-1))^(m*k+m-1): A262736 (m=2), A262948 (m=3), this sequence (m=4), A285340 (m=5).

Cf. A285213, A285311.

a(n) = (-1)^n * A285213(n).

a(n) ~ exp(3^(4/3) * Zeta(3)^(1/3) * n^(2/3) / 4) * Zeta(3)^(1/6) / (2^(23/24) * 3^(1/3) * sqrt(Pi) * n^(2/3)). - Vaclav Kotesovec, Nov 10 2017

A291647 Expansion of Product_{k>=1} (1 + x^prime(k))^prime(k).

Original entry on oeis.org

1, 0, 2, 3, 1, 11, 3, 20, 21, 20, 64, 35, 112, 117, 160, 269, 284, 477, 598, 819, 1116, 1495, 1899, 2718, 3389, 4596, 6121, 7627, 10460, 13128, 17350, 22506, 28696, 37063, 47779, 60249, 78642, 98783, 126058, 160758, 200795, 257750, 321768, 407930, 511526, 640636, 802816, 1005618, 1252820, 1567454, 1946162

Offset: 0

Ilya Gutkovskiy, Aug 28 2017

nonn

Number of partitions of n into distinct prime parts, where prime(k) different parts of size prime(k) are available (2a, 2b, 3a, 3b, 3c, ...).

			a(6) = 3 because we have [3a, 3b], [3a, 3c] and [3b, 3c].

Index entries for related partition-counting sequences

Cf. A000040, A000586, A007441, A026007, A061151, A061152, A219224, A262736.

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

G.f.: Product_{k>=1} (1 + x^A000040(k))^A000040(k).

cp's OEIS Frontend

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

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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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A285338 Expansion of Product_{k>=1} (1 + x^(5*k-4))^(5*k-4).

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A285340 Expansion of Product_{k>=0} (1 + x^(5*k+4))^(5*k+4).

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A285339 Expansion of Product_{k>=0} (1 + x^(4*k+3))^(4*k+3).

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A291647 Expansion of Product_{k>=1} (1 + x^prime(k))^prime(k).

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

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

A285338 Expansion of Product_{k>=1} (1 + x^(5k-4))^(5k-4).

A285340 Expansion of Product_{k>=0} (1 + x^(5k+4))^(5k+4).

A285339 Expansion of Product_{k>=0} (1 + x^(4k+3))^(4k+3).