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 1-4 of 4 results.

A284474 Expansion of Product_{k>=1} (1 + x^(2*k))^(2*k)/(1 + x^(2*k-1))^(2*k-1).

Original entry on oeis.org

1, -1, 3, -6, 11, -22, 42, -74, 131, -231, 395, -669, 1122, -1851, 3029, -4915, 7891, -12572, 19881, -31203, 48657, -75391, 116096, -177792, 270822, -410394, 618905, -929052, 1388403, -2066140, 3062270, -4520912, 6649463, -9745072, 14232278, -20716355, 30057438
Offset: 0

Views

Author

Seiichi Manyama, Apr 15 2017

Keywords

Crossrefs

Programs

  • Mathematica
    nmax = 50; CoefficientList[Series[Product[(1 + x^(2*k))^(2*k)/(1 + x^(2*k-1))^(2*k-1), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Apr 15 2017 *)

Formula

a(n) ~ (-1)^n * exp(-1/12 + 3 * 2^(-5/3) * (7*Zeta(3))^(1/3) * n^(2/3)) * A * (7*Zeta(3))^(5/36) / (2^(10/9) * sqrt(3*Pi) * n^(23/36)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Apr 17 2017
G.f.: exp(Sum_{k>=1} (-1)^k*x^k/(k*(1 + x^k)^2)). - Ilya Gutkovskiy, Jun 20 2018

A281683 Expansion of Product_{k>=1} (1 - x^(2*k-1))^(2*k-1)/(1 - x^(2*k))^(2*k).

Original entry on oeis.org

1, -1, 2, -5, 10, -18, 32, -59, 106, -181, 305, -518, 867, -1418, 2301, -3724, 5966, -9448, 14862, -23263, 36165, -55802, 85609, -130732, 198574, -299941, 450946, -675153, 1006395, -1493598, 2207928, -3251926, 4771934, -6977018, 10166502, -14766512, 21379861
Offset: 0

Views

Author

Seiichi Manyama, Apr 14 2017

Keywords

Crossrefs

Programs

  • Mathematica
    nmax = 50; CoefficientList[Series[Product[(1 - x^k)^k/(1 - x^(2*k))^(4*k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 09 2017 *)
    nmax = 50; CoefficientList[Series[Product[1/((1 + x^k)^(4*k)*(1 - x^k)^(3*k)), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 09 2017 *)

Formula

a(n) = (-1)^n * A224364(n).
a(n) ~ (-1)^n * exp(1/6 + 3 * 2^(-5/3) * (7*Zeta(3))^(1/3) * n^(2/3)) * (7*Zeta(3))^(2/9) / (2^(25/36) * A^2 * sqrt(3*Pi) * n^(13/18)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Nov 09 2017

A307460 Expansion of Product_{k>=1} (1-x^k)^((-1)^k*k^2).

Original entry on oeis.org

1, 1, -3, 6, -4, -15, 54, -87, 63, 79, -405, 912, -1363, 1193, 510, -4900, 12512, -21582, 26512, -16540, -24585, 113682, -255045, 419931, -519210, 377176, 267957, -1703694, 4090424, -7179222, 9895981, -9897664, 3337614, 14790666, -49171217, 100903743
Offset: 0

Views

Author

Seiichi Manyama, Apr 09 2019

Keywords

Comments

This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = (-1)^(n+1) * n^2, g(n) = 1.

Crossrefs

Product_{k>=1} (1-x^k)^((-1)^k*k^b): A010054 (b=0), A281781 (b=1), this sequence (b=2).

Programs

  • Mathematica
    nmax = 40; CoefficientList[Series[Product[(1 - x^k)^((-1)^k*k^2), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Apr 09 2019 *)
  • PARI
    N=66; x='x+O('x^N); Vec(prod(k=1, N, (1-x^k)^((-1)^k*k^2)))

A307514 Expansion of Product_{k>=1} (1-x^k)^((-1)^k*k^k).

Original entry on oeis.org

1, 1, -3, 24, -226, 2791, -42467, 761826, -15714798, 366401751, -9528266885, 273439284005, -8584541521286, 292695692569785, -10771202678289501, 425538242701632216, -17964593967281888258, 807094224863059707077, -38449142619220645357810, 1935991142823285710574298
Offset: 0

Views

Author

Seiichi Manyama, Apr 12 2019

Keywords

Comments

This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = (-1)^(n+1) * n^n, g(n) = 1.

Crossrefs

Programs

  • Mathematica
    nmax=20; CoefficientList[Series[Product[(1-x^k)^((-1)^k*k^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Apr 12 2019 *)
  • PARI
    N=20; x='x+O('x^N); Vec(prod(k=1, N, (1-x^k)^((-1)^k*k^k)))

Formula

a(n) ~ -(-1)^n * n^n * (1 - exp(-1)/n - (exp(-1)/2 + 3*exp(-2))/n^2). - Vaclav Kotesovec, Apr 12 2019
Showing 1-4 of 4 results.