A279928 -id:A279928 - cp's OEIS frontend

A281266 Main diagonal of A279928.

1, -1, -1, -1, 23, -51, 35, -197, 1367, -3889, 7649, -26258, 112739, -350676, 939623, -3063201, 11022167, -35276497, 106320311, -344831533, 1164544273, -3765456206, 11890410454, -38631658591, 127610160227, -414671018176, 1335126443260, -4348160271568

Offset: 0

Seiichi Manyama, Apr 13 2017

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Vaclav Kotesovec, Table of n, a(n) for n = 0..1000 (terms 0..100 from Seiichi Manyama)

Cf. A255672, A279928.

nmax = 30; Table[SeriesCoefficient[Product[1/(1 + x^k)^(m*k), {k, 1, m}], {x, 0, m}], {m, 0, nmax}] (* Vaclav Kotesovec, Apr 17 2017 *)

a(n) ~ (-1)^n * c * d^n / sqrt(n), where d = 3.31585574856163070436... and c = 0.20250147602443379616... - Vaclav Kotesovec, Apr 17 2017

a(n) = [x^n] exp(n*Sum_{k>=1} (-1)^k*x^k/(k*(1 - x^k)^2)). - Ilya Gutkovskiy, May 30 2018

A277938 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1+x^j)^(j*k) in powers of x.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 5, 5, 0, 1, 4, 9, 14, 8, 0, 1, 5, 14, 28, 30, 16, 0, 1, 6, 20, 48, 72, 68, 28, 0, 1, 7, 27, 75, 141, 183, 145, 49, 0, 1, 8, 35, 110, 245, 396, 443, 298, 83, 0, 1, 9, 44, 154, 393, 751, 1058, 1026, 600, 142, 0, 1, 10, 54, 208

Offset: 0

Seiichi Manyama, Apr 11 2017

			Square array begins:
   1, 1,  1,  1,   1, ...
   0, 1,  2,  3,   4, ...
   0, 2,  5,  9,  14, ...
   0, 5, 14, 28,  48, ...
   0, 8, 30, 72, 141, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns k=0-4 give: A000007, A026007, A026011, A027346, A027906.
Rows n=0-3 give: A000012, A001477, A000096, A005586.
Main diagonal gives A270922.
Antidiagonal sums give A299167.
Cf. A276554, A279928.

G.f. of column k: Product_{j>=1} (1+x^j)^(j*k).

A279411 Expansion of Product_{k>0} 1/(1 + x^k)^(k*4).

Original entry on oeis.org

1, -4, 2, 0, 23, -20, 2, -88, 63, -96, 318, -104, 626, -844, 504, -2472, 1525, -3704, 6184, -4288, 15284, -10736, 23254, -35792, 30228, -84544, 60974, -139240, 176658, -190108, 418940, -320976, 755332, -773524, 1111678, -1847304, 1669046, -3634296

Offset: 0

Seiichi Manyama, Apr 11 2017

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

Column k=4 of A279928.

Product_{k>0} 1/(1 + x^k)^(k*m): A027906 (m=-4), A255528 (m=1), A278710 (m=2), A279031 (m=3), this sequence (m=4), A279932 (m=5).

a(n) ~ (-1)^n * exp(-1/3 + 3/2 * Zeta(3)^(1/3) * n^(2/3)) * A^4 * Zeta(3)^(1/18) / (sqrt(6*Pi) * n^(5/9)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Apr 13 2017

G.f.: exp(4*Sum_{k>=1} (-1)^k*x^k/(k*(1 - x^k)^2)). - Ilya Gutkovskiy, Mar 27 2018

A299212 Expansion of 1/(1 - x*Product_{k>=1} 1/(1 + x^k)^k).

Original entry on oeis.org

1, 1, 0, -2, -5, -4, 4, 21, 35, 23, -47, -165, -239, -78, 479, 1273, 1508, -138, -4429, -9451, -8845, 6207, 37937, 67123, 45144, -83355, -308078, -455109, -166872, 873799, 2393041, 2916869, -73472, -8133572, -17828640, -17294146, 10383571, 70275162, 127401305, 90368779, -147825714

Offset: 0

Ilya Gutkovskiy, Feb 05 2018

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N. J. A. Sloane, Transforms

Antidiagonal sums of A279928.

Cf. A067687, A255528, A299105, A299106, A299108, A299162, A299164, A299166, A299167, A299208, A299209, A299210, A299211.

nmax = 40; CoefficientList[Series[1/(1 - x Product[1/(1 + x^k)^k, {k, 1, nmax}]), {x, 0, nmax}], x]

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

a(0) = 1; a(n) = Sum_{k=1..n} A255528(k-1)*a(n-k).

A279932 Expansion of Product_{k>0} 1/(1 + x^k)^(k*5).

Original entry on oeis.org

1, -5, 5, 0, 30, -51, 5, -130, 220, -125, 649, -605, 870, -2695, 1565, -4852, 7915, -6360, 20625, -17880, 33551, -61015, 50865, -138510, 135485, -224725, 389025, -359610, 849525, -838970, 1417404, -2195205, 2275690, -4756040, 4657940, -8315123, 11174840, -13352315

Offset: 0

Seiichi Manyama, Apr 12 2017

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In general, if m >= 1 and g.f. = Product_{k>=1} 1/(1 + x^k)^(m*k), then a(n, m) ~ (-1)^n * exp(-m/12 + 3 * 2^(-5/3) * m^(1/3) * Zeta(3)^(1/3) * n^(2/3)) * 2^(m/18 - 5/6) * A^m * m^(1/6 - m/36) * Zeta(3)^(1/6 - m/36) * n^(m/36 - 2/3) / sqrt(3*Pi), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Apr 13 2017

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

Column k=5 of A279928.

Product_{k>0} 1/(1 + x^k)^(k*m): A027906 (m=-4), A255528 (m=1), A278710 (m=2), A279031 (m=3), A279411 (m=4), this sequence (m=5).

a(n) ~ (-1)^n * exp(-5/12 + 3 * 2^(-5/3) * (5*Zeta(3))^(1/3) * n^(2/3)) * A^5 * (5*Zeta(3))^(1/36) / (2^(5/9) * sqrt(3*Pi) * n^(19/36)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Apr 13 2017

G.f.: exp(5*Sum_{k>=1} (-1)^k*x^k/(k*(1 - x^k)^2)). - Ilya Gutkovskiy, Mar 27 2018

cp's OEIS Frontend

A281266 Main diagonal of A279928.

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A277938 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1+x^j)^(j*k) in powers of x.

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A279411 Expansion of Product_{k>0} 1/(1 + x^k)^(k*4).

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A299212 Expansion of 1/(1 - x*Product_{k>=1} 1/(1 + x^k)^k).

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A279932 Expansion of Product_{k>0} 1/(1 + x^k)^(k*5).

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