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 1-7 of 7 results.

A327688 Expansion of Product_{k>=1} B(x^k), where B(x) is the g.f. of A007325.

Original entry on oeis.org

1, -1, 0, 0, -1, 0, 1, 0, 0, 0, 0, -2, 2, 1, 0, 1, -1, -1, -1, -1, 2, 1, 0, 1, -1, -3, 1, 2, -1, 0, 4, -6, -2, 3, -1, 1, 4, -1, -2, -1, 2, -4, 4, 0, -3, 1, -3, 4, 2, -1, 3, -1, -3, -1, 2, -3, 1, 2, -6, -3, 12, -7, 3, 11, -7, -4, 7, -10, -1, 7, 2, -16, 11, 2, -10, 14, -4, 3, -3
Offset: 0

Views

Author

Seiichi Manyama, Sep 22 2019

Keywords

Links

Crossrefs

Cf. A007325, A035187, A307757, A327686, A327690, A327691, A327716.

Programs

  • PARI
    N=66; x='x+O('x^N); Vec(prod(k=1, N, (1-x^k)^sumdiv(k, d, kronecker(5, d))))

Formula

G.f.: Product_{i>=1} Product_{j>=1} (1-x^(i*(5*j-1))) * (1-x^(i*(5*j-4))) / ((1-x^(i*(5*j-2))) * (1-x^(i*(5*j-3)))).
G.f.: Product_{k>=1} (1-x^k)^A035187(k).

A327717 Expansion of Product_{k>=1} (1 + x^k/(1 + x^(2*k))).

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 2, 3, 5, 6, 6, 7, 10, 12, 12, 15, 20, 23, 24, 28, 36, 42, 44, 51, 64, 73, 78, 89, 108, 123, 132, 150, 179, 202, 218, 246, 288, 324, 350, 393, 456, 509, 552, 616, 706, 786, 852, 948, 1078, 1195, 1297, 1436, 1620, 1791, 1942, 2145, 2406, 2650, 2874, 3163, 3528
Offset: 0

Views

Author

Seiichi Manyama, Sep 23 2019

Keywords

Links

Crossrefs

Convolution inverse of A307757.
Cf. A000726, A327716, A327718, A327719, A327720.

Programs

  • Mathematica
    nmax = 100; CoefficientList[Series[Product[1 + x^k/(1 + x^(2*k)), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 23 2019 *)
nmax = 100; CoefficientList[Series[Product[(1 + x^k + x^(2*k)) * (1 - x^(4*k - 2)), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 23 2019 *)
  • PARI
    N=66; x='x+O('x^N); Vec(prod(k=1, N, 1+x^k/(1+x^(2*k))))

Formula

a(n) ~ 5^(1/4) * exp(sqrt(5*n/2)*Pi/3) / (2^(5/4)*3*n^(3/4)). - Vaclav Kotesovec, Sep 23 2019

A327718 Expansion of Product_{k>=1} (1 + x^k/(1 + x^(2*k)/(1 + x^(3*k)))).

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 3, 3, 4, 6, 8, 9, 9, 11, 15, 20, 21, 20, 24, 36, 48, 46, 41, 52, 80, 100, 88, 74, 103, 170, 207, 166, 124, 198, 354, 409, 269, 162, 369, 745, 802, 382, 136, 706, 1585, 1515, 328, -178, 1422, 3481, 2822, -387, -1283, 3144, 7816, 4951, -3451, -4472, 7694, 18055
Offset: 0

Views

Author

Seiichi Manyama, Sep 23 2019

Keywords

Links

Crossrefs

Convolution inverse of A327686.
Cf. A327716, A327717, A327719, A327720.

Programs

  • Mathematica
    nmax = 60; CoefficientList[Series[Product[(1 - x^(5*k)) / ((1 - x^k)*(1 + x^(2*k) + x^(3*k))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 23 2019 *)
  • PARI
    N=66; x='x+O('x^N); Vec(prod(k=1, N, 1+x^k/(1+x^(2*k)/(1+x^(3*k)))))

A327719 Expansion of Product_{k>=1} (1 + x^k/(1 + x^(2*k)/(1 + x^(3*k)/(1 + x^(4*k))))).

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 3, 3, 4, 6, 7, 9, 10, 12, 14, 16, 21, 24, 27, 32, 37, 45, 52, 59, 69, 76, 89, 103, 118, 137, 148, 173, 197, 225, 256, 280, 324, 362, 409, 462, 508, 579, 644, 720, 811, 892, 1006, 1114, 1243, 1389, 1519, 1701, 1882, 2090, 2316, 2538, 2825, 3110, 3437, 3795, 4153
Offset: 0

Views

Author

Seiichi Manyama, Sep 23 2019

Keywords

Links

Crossrefs

Cf. A327716, A327717, A327718, A327720.

Programs

  • Mathematica
    nmax = 60; CoefficientList[Series[Product[(1 + x^k + x^(2*k) + x^(3*k) + 2*x^(4*k) + x^(5*k) + x^(6*k)) / (1 + x^(2*k) + x^(3*k) + x^(4*k) + x^(6*k)), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 23 2019 *)
  • PARI
    N=66; x='x+O('x^N); Vec(prod(k=1, N, 1+x^k/(1+x^(2*k)/(1+x^(3*k)/(1+x^(4*k))))))

A327720 Expansion of Product_{k>=1} (1 + x^k/(1 + x^(2*k)/(1 + x^(3*k)/(1 + x^(4*k)/(1 + x^(5*k)))))).

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 3, 3, 4, 6, 7, 9, 10, 12, 14, 17, 21, 23, 26, 32, 40, 46, 51, 57, 68, 80, 91, 102, 115, 135, 156, 180, 197, 219, 250, 290, 333, 364, 403, 456, 526, 594, 648, 708, 803, 922, 1037, 1128, 1223, 1373, 1560, 1752, 1902, 2062, 2300, 2613, 2926, 3149, 3378, 3740
Offset: 0

Views

Author

Seiichi Manyama, Sep 23 2019

Keywords

Comments

a(169) = -3504817 < 0.

Links

Crossrefs

Cf. A327716, A327717, A327718, A327719.

Programs

  • Mathematica
    nmax = 60; CoefficientList[Series[Product[(1 + x^k/(1 + x^(2*k)/(1 + x^(3*k)/(1 + x^(4*k)/(1 + x^(5*k)))))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 23 2019 *)
  • PARI
    N=66; x='x+O('x^N); Vec(prod(k=1, N, 1+x^k/(1+x^(2*k)/(1+x^(3*k)/(1+x^(4*k)/(1+x^(5*k)))))))

A327851 Expansion of Product_{k>=1} B(x^k), where B(x) is the g.f. of A111374.

Original entry on oeis.org

1, 1, 2, 2, 4, 4, 6, 8, 12, 15, 19, 24, 30, 36, 47, 57, 74, 88, 112, 130, 160, 190, 232, 277, 333, 399, 471, 554, 656, 768, 908, 1060, 1256, 1452, 1702, 1968, 2294, 2646, 3068, 3549, 4093, 4710, 5418, 6211, 7121, 8138, 9331, 10625, 12150, 13817, 15749, 17858, 20290, 23000, 26054
Offset: 0

Views

Author

Seiichi Manyama, Sep 28 2019

Keywords

Comments

a(n) > 0.

Links

Crossrefs

Convolution inverse of A327852.
Product_{k>=1} (1 - x^k)^(- Sum_{d|k} (b/d)), where (m/n) is the Kronecker symbol: this sequence (b=2), A107742 (b=4), A327716 (b=5).
Cf. A035185, A111374.

Programs

  • Mathematica
    nmax = 60; CoefficientList[Series[Product[QPochhammer[x^(8*j - 3)] * QPochhammer[x^(8*j - 5)]/(QPochhammer[x^(8*j - 7)] * QPochhammer[x^(8*j - 1)]), {j, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 28 2019 *)
  • PARI
    N=66; x='x+O('x^N); Vec(1/prod(k=1, N, (1-x^k)^sumdiv(k, d, kronecker(2, d))))

Formula

G.f.: Product_{i>=1} Product_{j>=1} (1-x^(i*(8*j-3))) * (1-x^(i*(8*j-5))) / ((1-x^(i*(8*j-1))) * (1-x^(i*(8*j-7)))).
G.f.: Product_{k>=1} (1-x^k)^(-A035185(k)).

A327694 Expansion of Product_{k>=1} B(x^k), where B(x) is the g.f. of A284321.

Original entry on oeis.org

1, 0, -1, -1, -1, 1, -1, 0, 0, 1, 4, 2, -1, -2, 1, 2, 0, -5, -2, 0, 5, -1, -6, -7, -3, 6, -1, -2, -6, 7, 18, 7, -8, -6, 1, 12, 4, -10, -7, 6, 27, 10, -21, -25, -1, 19, -4, -29, -26, 11, 39, 6, -27, -42, -3, 40, 12, -45, -32, 28, 90, 24, -55, -57, 9, 75, 14, -57, -59, 57, 139
Offset: 0

Views

Author

Seiichi Manyama, Sep 23 2019

Keywords

Crossrefs

Convolution inverse of A327691.
Cf. A284321, A327716.

Formula

G.f.: Product_{i>=1} Product_{j>=1} (1 - x^(i*(5*j-2)))*(1 - x^(i*(5*j-3))).
Showing 1-7 of 7 results.

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

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