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 1-10 of 13 results. Next

A285070 Expansion of Product_{k>=0} (1-x^(4*k+1))^(4*k+1).

Original entry on oeis.org

1, -1, 0, 0, 0, -5, 5, 0, 0, -9, 19, -10, 0, -13, 58, -55, 10, -17, 118, -191, 95, -26, 223, -512, 400, -116, 362, -1175, 1329, -564, 609, -2368, 3593, -2218, 1246, -4402, 8600, -7118, 3433, -7792, 18503, -19778, 10702, -13924, 37009, -49017, 32097, -27141
Offset: 0

Views

Author

Seiichi Manyama, Apr 15 2017

Keywords

Links

Crossrefs

Product_{k>=0} (1-x^(m*k+1))^(m*k+1): A285069 (m=2), A285050 (m=3), this sequence (m=4), A285071 (m=5).
Cf. A285048, A285288.

Programs

  • Mathematica
    nmax = 50; CoefficientList[Series[Product[(1-x^(4*k-3))^(4*k-3), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Apr 17 2017 *)

Formula

a(n) ~ (-1)^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, Apr 17 2017

A285071 Expansion of Product_{k>=0} (1-x^(5*k+1))^(5*k+1).

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, -25, 538, -2043, 2681, -1310, 130, 843, -4184, 7426, -5390, 1365, 1158, -7855, 18067, -17705, 7185, 798, -13701, 39468, -50030
Offset: 0

Views

Author

Seiichi Manyama, Apr 15 2017

Keywords

Links

Crossrefs

Product_{k>=0} (1-x^(m*k+1))^(m*k+1): A285069 (m=2), A285050 (m=3), A285070 (m=4), this sequence (m=5).
Cf. A285049, A285338.

Programs

  • Mathematica
    nmax = 50; CoefficientList[Series[Product[(1-x^(5*k-4))^(5*k-4), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Apr 17 2017 *)

A285213 Expansion of Product_{k>=0} (1-x^(4*k+3))^(4*k+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
Offset: 0

Views

Author

Seiichi Manyama, Apr 15 2017

Keywords

Links

Crossrefs

Product_{k>=0} (1-x^(m*k+m-1))^(m*k+m-1): A285069 (m=2), A285212 (m=3), this sequence (m=4), A285214 (m=5).
Cf. A285131.

Programs

  • Mathematica
    nmax = 50; CoefficientList[Series[Product[(1-x^(4*k-1))^(4*k-1), {k,1,nmax}], {x,0,nmax}], x] (* Vaclav Kotesovec, Apr 15 2017 *)
  • PARI
    x='x+O('x^100); Vec(prod(k=0, 100, (1 - x^(4*k + 3))^(4*k + 3))) \\ Indranil Ghosh, Apr 15 2017

Formula

a(n) ~ (-1)^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, Apr 17 2017

A285214 Expansion of Product_{k>=0} (1-x^(5*k+4))^(5*k+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, 2, -1027, 3810, -3605, 904, 423, -4600, 10196, -6898, 1240, 2990, -15805, 24104, -12242, 822, 14005, -46090, 51376
Offset: 0

Views

Author

Seiichi Manyama, Apr 15 2017

Keywords

Links

Crossrefs

Product_{k>=0} (1-x^(m*k+m-1))^(m*k+m-1): A285069 (m=2), A285212 (m=3), A285213 (m=4), this sequence (m=5).
Cf. A285132.

Programs

  • Mathematica
    nmax = 100; CoefficientList[Series[Product[(1-x^(5*k+4))^(5*k+4), {k,0,nmax}], {x,0,nmax}], x] (* Vaclav Kotesovec, Apr 15 2017 *)
  • PARI
    x='x+O('x^100); Vec(prod(k=0, 100, (1 - x^(5*k + 4))^(5*k + 4))) \\ Indranil Ghosh, Apr 15 2017

A285050 Expansion of Product_{k>=0} (1-x^(3*k+1))^(3*k+1).

Original entry on oeis.org

1, -1, 0, 0, -4, 4, 0, -7, 13, -6, -10, 38, -32, -9, 74, -103, 27, 137, -266, 153, 191, -593, 537, 167, -1161, 1437, -222, -2035, 3397, -1578, -3110, 7160, -5285, -3712, 13942, -13920, -2002, 24848, -32241, 6764, 40661, -68059, 32487, 59109, -133506, 95221, 71243
Offset: 0

Views

Author

Seiichi Manyama, Apr 15 2017

Keywords

Links

Crossrefs

Product_{k>=0} (1-x^(m*k+1))^(m*k+1): A285069 (m=2), this sequence (m=3), A285070 (m=4), A285071 (m=5).
Cf. A262947.

A285212 Expansion of Product_{k>=0} (1-x^(3*k+2))^(3*k+2).

Original entry on oeis.org

1, 0, -2, 0, 1, -5, 0, 10, -8, -5, 26, -11, -28, 62, -4, -101, 111, 43, -260, 182, 228, -583, 202, 715, -1155, 25, 1888, -2034, -851, 4286, -3144, -3418, 8895, -3888, -9806, 16848, -2479, -23812, 29519, 5626, -52156, 46930, 30033, -105320, 66001, 90431, -198736
Offset: 0

Views

Author

Seiichi Manyama, Apr 15 2017

Keywords

Links

Crossrefs

Product_{k>=0} (1-x^(m*k+m-1))^(m*k+m-1): A285069 (m=2), this sequence (m=3), A285213 (m=4), A285214 (m=5).
Cf. A262946.

Programs

  • Mathematica
    nmax = 100; CoefficientList[Series[Product[(1-x^(3*k+2))^(3*k+2), {k,0,nmax}], {x,0,nmax}], x] (* Vaclav Kotesovec, Apr 15 2017 *)
  • PARI
    x='x+O('x^100); Vec(prod(k=0, 100, (1 - x^(3*k + 2))^(3*k + 2))) \\ Indranil Ghosh, Apr 15 2017

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

Views

Author

Vaclav Kotesovec, Sep 08 2017

Keywords

Comments

Convolution of A262736 and A262811.

Links

Crossrefs

Cf. A080054, A262736, A262811, A284628, A285069, A292038.

Programs

  • Mathematica
    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]

Formula

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

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

Links

Crossrefs

Cf. A224364, A285069, A281781.

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

A285284 Expansion of Product_{k>=1} (1 - x^k)^k/(1 - x^(4*k))^(4*k).

Original entry on oeis.org

1, -1, -2, -1, 4, 0, -4, 3, 21, -4, -29, -7, 51, -24, -105, -7, 201, -30, -291, 34, 642, -42, -874, 75, 1764, -262, -2737, -40, 4555, -818, -7512, 88, 12425, -1492, -19062, 1135, 32637, -2573, -47688, 3576, 81335, -6477, -119540, 6525, 193738, -18478, -292685
Offset: 0

Views

Author

Seiichi Manyama, Apr 16 2017

Keywords

Links

Crossrefs

Product_{k>=1} (1 - x^k)^k/(1 - x^(m*k))^(m*k): A285069 (m=2), A285247 (m=3), this sequence (m=4), A285285 (m=5).
Cf. A285215, A285292.

Programs

  • Mathematica
    nmax = 50; CoefficientList[Series[Product[(1 - x^k)^k/(1 - x^(4*k))^(4*k), {k,1,nmax}], {x,0,nmax}], x] (* Vaclav Kotesovec, Apr 16 2017 *)

A285285 Expansion of Product_{k>=1} (1 - x^k)^k/(1 - x^(5*k))^(5*k).

Original entry on oeis.org

1, -1, -2, -1, 0, 9, -1, -3, -2, -2, 36, -22, -40, -34, -23, 154, -53, -89, -48, -4, 652, -161, -352, -293, -213, 1974, -813, -1416, -1198, -838, 6386, -2328, -3970, -3054, -1682, 20641, -5637, -10436, -8183, -4723, 59693, -17856, -32830, -28002, -19303
Offset: 0

Views

Author

Seiichi Manyama, Apr 16 2017

Keywords

Links

Crossrefs

Product_{k>=1} (1 - x^k)^k/(1 - x^(m*k))^(m*k): A285069 (m=2), A285247 (m=3), A285284 (m=4), this sequence (m=5).

Programs

  • Mathematica
    nmax = 50; CoefficientList[Series[Product[(1 - x^k)^k/(1 - x^(5*k))^(5*k), {k,1,nmax}], {x,0,nmax}], x] (* Vaclav Kotesovec, Apr 16 2017 *)
Showing 1-10 of 13 results. Next

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