cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A255528 G.f.: Product_{k>=1} 1/(1+x^k)^k.

Original entry on oeis.org

1, -1, -1, -2, 1, 0, 4, 2, 8, -2, 4, -11, -1, -25, -5, -35, 13, -26, 49, -6, 110, 6, 159, -23, 182, -141, 129, -358, 62, -640, 39, -897, 237, -1013, 771, -914, 1793, -664, 3143, -565, 4635, -1157, 5727, -3119, 6121, -7041, 5642, -13088, 5097, -20758, 5879
Offset: 0

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Author

Vaclav Kotesovec, Feb 24 2015

Keywords

Comments

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

Links

Crossrefs

Cf. A000219, A026007, A026011, A073592, A262736.
Cf. A278710 (m=2), A279031 (m=3), A279411 (m=4), A279932 (m=5).

Programs

  • Maple
    with(numtheory): A000219:=proc(n) option remember; if n = 0 then 1 else add(sigma[2](k)*A000219(n-k), k = 1..n)/n fi: end: A073592:=proc(n) option remember; if n = 0 then 1 else -add(sigma[2](k)*A073592(n-k), k = 1..n)/n fi: end: a:=proc(n); add(A073592(n-2*m)*A000219(m), m = 0..floor(n/2)): end: seq(a(n), n = 0..50); # Vaclav Kotesovec, Mar 09 2015
  • Mathematica
    nmax=100; CoefficientList[Series[Product[1/(1+x^k)^k,{k,1,nmax}],{x,0,nmax}],x]
  • PARI
    {a(n) = if(n<0, 0, polcoeff(exp(sum(k=1, n, (-1)^k * x^k / (1-x^k)^2 / k, x*O(x^n))), n))}
for(n=0, 100, print1(a(n), ", "))

Formula

a(n) ~ (-1)^n * A * Zeta(3)^(5/36) * exp(3*Zeta(3)^(1/3)*n^(2/3)/2^(5/3) - 1/12) / (2^(7/9) * sqrt(3*Pi) * n^(23/36)), where Zeta(3) = A002117 and A = A074962 is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Sep 29 2015
a(0) = 1, a(n) = -(1/n)*Sum_{k=1..n} A078306(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 06 2017

A278710 Convolution square of A255528.

Original entry on oeis.org

1, -2, -1, -2, 7, 2, 10, -8, 5, -40, -4, -54, 52, -30, 162, -12, 292, -142, 270, -576, 168, -1228, 305, -1702, 1435, -1664, 3839, -1444, 7303, -2752, 10117, -8420, 11065, -20714, 11066, -38702, 17057, -57276, 40310, -69898, 94138, -77014, 181926, -97480
Offset: 0

Views

Author

Seiichi Manyama, Apr 11 2017

Keywords

Links

Crossrefs

Product_{k>0} 1/(1 + x^k)^(k*m): A026011 (m=-2), A255528 (m=1), this sequence (m=2), A279031 (m=3), A279411 (m=4).

Formula

G.f.: Product_{k>0} 1/(1 + x^k)^(k*2).
a(n) ~ (-1)^n * exp(-1/6 + 3 * 2^(-4/3) * Zeta(3)^(1/3) * n^(2/3)) * A^2 * Zeta(3)^(1/9) / (2^(11/18) * sqrt(3*Pi) * n^(11/18)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Apr 13 2017
G.f.: exp(2*Sum_{k>=1} (-1)^k*x^k/(k*(1 - x^k)^2)). - Ilya Gutkovskiy, Mar 27 2018

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

Views

Author

Seiichi Manyama, Apr 11 2017

Keywords

Links

Crossrefs

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).

Formula

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

A279928 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} 1/(1+x^j)^(j*k) in powers of x.

Original entry on oeis.org

1, 1, 0, 1, -1, 0, 1, -2, -1, 0, 1, -3, -1, -2, 0, 1, -4, 0, -2, 1, 0, 1, -5, 2, -1, 7, 0, 0, 1, -6, 5, 0, 15, 2, 4, 0, 1, -7, 9, 0, 23, -3, 10, 2, 0, 1, -8, 14, -2, 30, -20, 8, -8, 8, 0, 1, -9, 20, -7, 36, -51, 2, -42, 5, -2, 0, 1, -10, 27, -16, 42, -96, 5, -88, 6
Offset: 0

Views

Author

Seiichi Manyama, Apr 11 2017

Keywords

Examples

			Square array begins:
   1,  1,  1,  1,  1, ...
   0, -1, -2, -3, -4, ...
   0, -1, -1,  0,  2, ...
   0, -2, -2, -1,  0, ...
   0,  1,  7, 15, 23, ...

Links

Crossrefs

Columns k=0-5 give: A000007, A255528, A278710, A279031, A279411, A279932.
Main diagonal gives A281266.
Antidiagonal sums give A299212.
Cf. A255961, A277938.

Formula

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

Views

Author

Seiichi Manyama, Apr 12 2017

Keywords

Comments

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

Links

Crossrefs

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).

Formula

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
Showing 1-5 of 5 results.

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