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 1-3 of 3 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

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)))))
Showing 1-3 of 3 results.

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

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