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.

A261451 Expansion of Product_{k>=1} ((1+x^k)/(1-x^k))^(k+1).

Original entry on oeis.org

1, 4, 14, 44, 124, 328, 824, 1980, 4590, 10320, 22584, 48268, 101016, 207432, 418704, 832032, 1629764, 3150280, 6014998, 11354084, 21204488, 39206168, 71811256, 130369900, 234704360, 419195412, 743085912, 1307823672, 2286094704, 3970174648, 6852048368
Offset: 0

Views

Author

Vaclav Kotesovec, Aug 19 2015

Keywords

Comments

Convolution of A005380 and A219555.

Links

Crossrefs

Cf. A156616, A005309, A261386, A261452, A261389.

Programs

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

Formula

a(n) ~ (7*Zeta(3))^(13/36) * exp(1/12 - Pi^4/(336*Zeta(3)) + Pi^2 * n^(1/3) / (2^(5/3) * (7*Zeta(3))^(1/3)) + 3/2 * ((7*Zeta(3))/2)^(1/3) * n^(2/3)) / (A * 2^(35/18) * 3^(1/2) * Pi * n^(31/36)), where Zeta(3) = A002117 and A = A074962 is the Glaisher-Kinkelin constant.

A005309 Fermionic string states.

Original entry on oeis.org

1, 0, 2, 4, 8, 16, 32, 60, 114, 212, 384, 692, 1232, 2160, 3760, 6480, 11056, 18728, 31474, 52492, 86976, 143176, 234224, 380988, 616288, 991624, 1587600, 2529560, 4011808, 6334656, 9960080, 15596532, 24327122, 37801568, 58525152, 90291232, 138825416
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

See the reference for precise definition.
The g.f. -(1-2*z+2*z**2)/(-1+2*z) conjectured by Simon Plouffe in his 1992 dissertation is not correct.

References

  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A156616, A261451, A261386, A261452, A261389.

Formula

G.f. Product_{k>=1} ((1+x^k)/(1-x^k))^(k-1). - Vaclav Kotesovec, Aug 19 2015
Convolution of A052847 and A052812. - Vaclav Kotesovec, Aug 19 2015
a(n) ~ 2^(7/18) * (7*Zeta(3))^(1/36) * exp(1/12 - Pi^4/(336*Zeta(3)) - Pi^2 * n^(1/3) / (2^(5/3)*(7*Zeta(3))^(1/3)) + 3/2 * (7*Zeta(3)/2)^(1/3) * n^(2/3)) / (A * sqrt(3) * n^(19/36)), where Zeta(3) = A002117 and A = A074962 is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Aug 19 2015

A302237 Expansion of Product_{k>=1} ((1 + x^k)/(1 - x^k))^(k*(k+1)/2).

Original entry on oeis.org

1, 2, 8, 26, 76, 216, 590, 1554, 3988, 9988, 24464, 58794, 138866, 322808, 739658, 1672372, 3734848, 8245956, 18012114, 38952586, 83448832, 177194716, 373111970, 779430870, 1615995262, 3326484686, 6800794428, 13813260736, 27881653590, 55942340000, 111601021856
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 03 2018

Keywords

Comments

Convolution of the sequences A000294 and A028377.

Links

Crossrefs

Cf. A000217, A000294, A015128, A028377, A156616, A206622, A206623, A206624, A260916, A261386, A261452, A261519, A261520, A301554, A301555.

Programs

  • Mathematica
    nmax = 30; CoefficientList[Series[Product[((1 + x^k)/(1 - x^k))^(k (k + 1)/2), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

G.f.: Product_{k>=1} ((1 + x^k)/(1 - x^k))^A000217(k).
a(n) ~ exp(2*Pi*n^(3/4)/3 + 7*Zeta(3)*sqrt(n) / (2*Pi^2) - 49*Zeta(3)^2 * n^(1/4) / (4*Pi^5) + 22411 * Zeta(3)^3 / (392*Pi^8) - Zeta(3)/(8*Pi^2) + 1/24) * Pi^(1/24) / (sqrt(A) * 2^(25/12) * n^(61/96)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Apr 08 2018
G.f.: A(x) = exp( 2*Sum_{n >= 0} x^(2*n+1)/((2*n+1)*(1 - x^(2*n+1))^3) ). Cf. A000122 and A156616. - Peter Bala, Dec 23 2021

A302238 Expansion of Product_{k>=1} ((1 + x^k)/(1 - x^k))^prime(k).

Original entry on oeis.org

1, 4, 14, 46, 136, 382, 1022, 2626, 6530, 15784, 37218, 85842, 194146, 431358, 943038, 2031454, 4316884, 9058662, 18787730, 38542526, 78264298, 157403290, 313712482, 619919350, 1215125262, 2363570168, 4563951858, 8751621598, 16670498062, 31553539214, 59361428202
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 03 2018

Keywords

Comments

Convolution of the sequences A030009 and A061152.

Links

Crossrefs

Cf. A000040, A015128, A030009, A061152, A156616, A206622, A206623, A206624, A260916, A261386, A261452, A261519, A261520, A301554, A301555.

Programs

  • Mathematica
    nmax = 30; CoefficientList[Series[Product[((1 + x^k)/(1 - x^k))^Prime[k], {k, 1, nmax}], {x, 0, nmax}], x]

Formula

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

A302239 Expansion of Product_{k>=1} ((1 + x^k)/(1 - x^k))^p(k), where p(k) = number of partitions of k (A000041).

Original entry on oeis.org

1, 2, 6, 16, 40, 96, 226, 512, 1140, 2488, 5336, 11270, 23494, 48356, 98438, 198338, 395846, 783136, 1536800, 2992818, 5786952, 11114950, 21213906, 40247696, 75928804, 142475644, 265985628, 494155176, 913802164, 1682338192, 3084101744, 5630853218, 10240484332, 18553818210
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 03 2018

Keywords

Comments

Convolution of the sequences A001970 and A261049.

Links

Crossrefs

Cf. A000041, A001970, A015128, A156616, A206622, A206623, A206624, A260916, A261049, A261386, A261452, A261519, A261520, A301554, A301555, A302237, A302238.

Programs

  • Mathematica
    nmax = 33; CoefficientList[Series[Product[((1 + x^k)/(1 - x^k))^PartitionsP[k], {k, 1, nmax}], {x, 0, nmax}], x]

Formula

G.f.: Product_{k>=1} ((1 + x^k)/(1 - x^k))^A000041(k).
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.