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 1-6 of 6 results.

A327716 Expansion of Product_{k>=1} B(x^k), where B(x) is the g.f. of A003823.

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, 45, 51, 58, 69, 80, 89, 102, 116, 135, 154, 177, 198, 224, 253, 288, 326, 361, 408, 459, 521, 583, 650, 723, 812, 909, 1009, 1122, 1244, 1393, 1547, 1716, 1898, 2101, 2326, 2575, 2845, 3132, 3456, 3809
Offset: 0

Views

Author

Seiichi Manyama, Sep 23 2019

Keywords

Comments

a(n) > 0.

Links

Crossrefs

Convolution inverse of A327688.
Cf. A003823, A035187, A327690, A327691, A327694, A327717, A327718, A327719, A327720.

Programs

  • Mathematica
    nmax = 60; CoefficientList[Series[Product[QPochhammer[x^(5*j - 3)] * QPochhammer[x^(5*j - 2)]/(QPochhammer[x^(5*j - 4)] * QPochhammer[x^(5*j - 1)]), {j, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 23 2019 *)
  • PARI
    N=66; x='x+O('x^N); Vec(1/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-2))) * (1-x^(i*(5*j-3))) / ((1-x^(i*(5*j-1))) * (1-x^(i*(5*j-4)))).
G.f.: Product_{k>=1} (1-x^k)^(-A035187(k)).
a(n) ~ c * exp(Pi*sqrt(r*n)) / n^(3/4), where r = 4*log((1+sqrt(5))/2) / (3*sqrt(5)) = 0.2869392939760026925..., c = 0.203427046022096... - Vaclav Kotesovec, Sep 24 2019, updated May 09 2020

A327691 Expansion of Product_{k>=1} B(x^k), where B(x) is the g.f. of A003106.

Original entry on oeis.org

1, 0, 1, 1, 2, 1, 5, 3, 8, 7, 13, 11, 26, 20, 40, 39, 66, 61, 111, 102, 171, 174, 266, 269, 427, 423, 638, 675, 969, 1016, 1477, 1544, 2177, 2350, 3209, 3466, 4754, 5112, 6867, 7546, 9931, 10899, 14343, 15729, 20406, 22653, 28962, 32168, 41069, 45561, 57551, 64382, 80491, 90030, 112286
Offset: 0

Views

Author

Seiichi Manyama, Sep 22 2019

Keywords

Links

Crossrefs

Cf. A003106, A327688, A327690.

Programs

  • Mathematica
    nmax = 50; CoefficientList[Series[Product[1/(QPochhammer[x^(5*j - 3)] * QPochhammer[x^(5*j - 2)]), {j, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 23 2019 *)

Formula

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

A327690 Expansion of Product_{k>=1} B(x^k), where B(x) is the g.f. of A003114.

Original entry on oeis.org

1, 1, 2, 3, 6, 8, 14, 19, 31, 43, 64, 88, 131, 176, 250, 337, 471, 626, 859, 1133, 1532, 2008, 2674, 3479, 4595, 5933, 7745, 9952, 12888, 16451, 21142, 26842, 34260, 43283, 54878, 68993, 87017, 108884, 136564, 170191, 212441, 263646, 327616, 405034, 501203, 617423, 760964
Offset: 0

Views

Author

Seiichi Manyama, Sep 22 2019

Keywords

Links

Crossrefs

Cf. A003114, A327688, A327691.

Programs

  • Mathematica
    nmax = 50; CoefficientList[Series[Product[1/(QPochhammer[x^(5*j - 4)] * QPochhammer[x^(5*j - 1)]), {j, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 23 2019 *)

Formula

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

A307757 Expansion of Product_{k>=1} 1/(1 + x^k/(1 + x^(2*k))).

Original entry on oeis.org

1, -1, 0, 0, -1, 0, 2, -2, 0, 2, -2, 0, 4, -4, 0, 4, -5, 0, 8, -8, 0, 8, -10, 0, 14, -15, 0, 16, -18, 0, 24, -26, 0, 28, -32, 0, 42, -44, 0, 48, -54, 0, 68, -72, 0, 80, -88, 0, 108, -115, 0, 128, -140, 0, 170, -180, 0, 200, -218, 0, 260, -276, 0, 308, -333, 0, 392, -416, 0, 464
Offset: 0

Views

Author

Seiichi Manyama, Sep 22 2019

Keywords

Crossrefs

Cf. A309733, A327686, A327688.

Programs

  • Mathematica
    m = 70; CoefficientList[Series[Product[1/(1 + x^k/(1 + x^(2*k))), {k, 1, m}], {x, 0, m}], x] (* Amiram Eldar, May 07 2021 *)
  • PARI
    N=66; x='x+O('x^N); Vec(1/prod(k=1, N, 1+x^k/(1+x^(2*k))))

A327686 Expansion of Product_{k>=1} 1/(1 + x^k/(1 + x^(2*k)/(1 + x^(3*k)))).

Original entry on oeis.org

1, -1, 0, 0, -1, 0, 1, 0, 0, 0, -1, 0, 2, 0, 0, -1, 0, 1, 0, -2, -2, 4, 1, 0, 1, -6, 2, 3, -2, -2, -2, 2, 3, 2, -1, -3, 3, -1, -1, 0, -7, 3, 7, -4, 0, 1, -1, 3, 4, -5, -3, 5, -3, -3, 5, -3, 0, 3, -8, -3, 11, -3, 1, 7, -5, 12, -2, -17, -3, 5, 9, -12, 6, -12, -1, 34, -26
Offset: 0

Views

Author

Seiichi Manyama, Sep 22 2019

Keywords

Crossrefs

Cf. A307757, A327688.

Programs

  • Mathematica
    m = 76; CoefficientList[Series[Product[1/(1 + x^k/(1 + x^(2*k)/(1 + x^(3*k)))), {k, 1, m}], {x, 0, m}], x] (* Amiram Eldar, May 06 2021 *)
  • PARI
    N=66; x='x+O('x^N); Vec(1/prod(k=1, N, 1+x^k/(1+x^(2*k)/(1+x^(3*k)))))

A327852 Expansion of Product_{k>=1} B(x^k), where B(x) is the g.f. of A092869.

Original entry on oeis.org

1, -1, -1, 1, -1, 1, 1, -3, 1, 2, 0, 2, -2, -2, -1, 3, 1, -5, 2, 0, 0, 8, -4, -7, 5, -2, 0, 1, -8, 0, 12, 2, -3, -1, -7, 9, 4, -7, -7, -6, 10, 9, 2, -6, -14, 15, 3, -15, 19, -30, 6, 37, -31, 10, 9, -23, 20, 4, -29, 4, 14, 4, -13, 23, -14, -19, 39, -29, -23, 35, 0, -34, 48
Offset: 0

Views

Author

Seiichi Manyama, Sep 28 2019

Keywords

Links

Crossrefs

Product_{k>=1} (1 - x^k)^(Sum_{d|k} (b/d)), where (m/n) is the Kronecker symbol: this sequence (b=2), A288007 (b=4), A327688 (b=5).
Cf. A035185, A092869, A327851.

Programs

  • PARI
    N=66; x='x+O('x^N); Vec(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-1))) * (1-x^(i*(8*j-7))) / ((1-x^(i*(8*j-3))) * (1-x^(i*(8*j-5)))).
G.f.: Product_{k>=1} (1-x^k)^A035185(k).
Showing 1-6 of 6 results.

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