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 1-5 of 5 results.

A374065 Expansion of Product_{k>=1} 1 / (1 + x^(3*k-2)).

Original entry on oeis.org

1, -1, 1, -1, 0, 0, 0, -1, 2, -2, 1, 0, -1, 0, 2, -3, 3, -1, -1, 1, 1, -4, 5, -3, 0, 2, 0, -4, 7, -6, 1, 3, -2, -3, 9, -10, 4, 3, -5, -1, 11, -15, 10, 1, -8, 3, 10, -20, 17, -3, -10, 9, 7, -24, 26, -10, -10, 15, 2, -27, 37, -21, -8, 22, -6, -28, 49, -36, -2, 30, -19, -24, 61, -56, 10, 35
Offset: 0

Views

Author

Ilya Gutkovskiy, Jun 27 2024

Keywords

Crossrefs

Cf. A035382, A081360, A081362, A109389, A132462, A261612, A262928, A284312, A374064.

Programs

  • Mathematica
    nmax = 75; CoefficientList[Series[Product[1/(1 + x^(3 k - 2)), {k, 1, nmax}], {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = (1/n) Sum[DivisorSum[k, (-1)^(k/#) # &, Mod[#, 3] == 1 &] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 75}]

Formula

a(0) = 1; a(n) = -Sum_{k=1..n} A261612(k) * a(n-k).
a(n) = Sum_{k=0..n} A081362(k) * A132462(n-k).
a(n) = Sum_{k=0..n} A109389(k) * A262928(n-k).

A374076 Expansion of Product_{k>=1} 1 / (1 + x^(5*k-1)).

Original entry on oeis.org

1, 0, 0, 0, -1, 0, 0, 0, 1, -1, 0, 0, -1, 1, -1, 0, 1, -1, 2, -1, -1, 1, -2, 2, 0, -1, 2, -3, 2, 0, -2, 3, -3, 2, 1, -3, 4, -4, 2, 2, -4, 5, -5, 1, 3, -6, 7, -5, 1, 5, -8, 8, -6, -1, 8, -10, 11, -6, -3, 10, -14, 12, -5, -6, 15, -17, 14, -4, -10, 19, -21, 15, -1, -15, 25, -25
Offset: 0

Views

Author

Ilya Gutkovskiy, Jun 27 2024

Keywords

Crossrefs

Cf. A081360, A081362, A109700, A281243, A284317, A374064, A374065, A374077, A374078, A374079.

Programs

  • Mathematica
    nmax = 75; CoefficientList[Series[Product[1/(1 + x^(5 k - 1)), {k, 1, nmax}], {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = (1/n) Sum[DivisorSum[k, (-1)^(k/#) # &, Mod[#, 5] == 4 &] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 75}]

A374077 Expansion of Product_{k>=1} 1 / (1 + x^(5*k-2)).

Original entry on oeis.org

1, 0, 0, -1, 0, 0, 1, 0, -1, -1, 0, 1, 1, -1, -1, -1, 2, 1, 0, -2, -1, 1, 2, 0, -2, -2, 2, 2, 1, -3, -2, 1, 4, 1, -3, -4, 2, 4, 3, -5, -5, 0, 7, 4, -4, -8, 0, 7, 8, -5, -9, -4, 10, 9, -3, -13, -5, 9, 14, -3, -14, -10, 12, 16, 1, -19, -12, 10, 23, 1, -20, -20, 13, 26, 8, -26
Offset: 0

Views

Author

Ilya Gutkovskiy, Jun 27 2024

Keywords

Crossrefs

Cf. A081360, A081362, A109699, A281271, A284320, A374064, A374065, A374076, A374078, A374079.

Programs

  • Mathematica
    nmax = 75; CoefficientList[Series[Product[1/(1 + x^(5 k - 2)), {k, 1, nmax}], {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = (1/n) Sum[DivisorSum[k, (-1)^(k/#) # &, Mod[#, 5] == 3 &] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 75}]

A374078 Expansion of Product_{k>=1} 1 / (1 + x^(5*k-3)).

Original entry on oeis.org

1, 0, -1, 0, 1, 0, -1, -1, 1, 1, -1, -1, 0, 1, 1, -1, -1, 0, 1, 1, -1, -2, 0, 2, 2, -2, -3, 1, 4, 1, -4, -3, 3, 4, 0, -5, -3, 4, 5, -1, -6, -3, 6, 6, -2, -8, -3, 8, 8, -5, -11, -2, 12, 8, -8, -13, 1, 15, 8, -12, -15, 3, 19, 7, -16, -17, 6, 23, 8, -22, -20, 11, 30, 5, -30, -22
Offset: 0

Views

Author

Ilya Gutkovskiy, Jun 27 2024

Keywords

Crossrefs

Cf. A081360, A081362, A109698, A281272, A284319, A374064, A374065, A374076, A374077, A374079.

Programs

  • Mathematica
    nmax = 75; CoefficientList[Series[Product[1/(1 + x^(5 k - 3)), {k, 1, nmax}], {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = (1/n) Sum[DivisorSum[k, (-1)^(k/#) # &, Mod[#, 5] == 2 &] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 75}]

A374079 Expansion of Product_{k>=1} 1 / (1 + x^(5*k-4)).

Original entry on oeis.org

1, -1, 1, -1, 1, -1, 0, 0, 0, 0, 0, -1, 2, -2, 2, -2, 1, 0, -1, 1, -1, 0, 2, -3, 4, -4, 3, -1, -1, 2, -3, 2, 1, -4, 6, -7, 7, -4, 0, 3, -5, 5, -2, -3, 8, -11, 12, -9, 3, 3, -8, 10, -7, 0, 8, -15, 19, -17, 9, 1, -10, 16, -15, 6, 7, -19, 28, -29, 20, -5, -11, 23, -26, 17, 1, -21
Offset: 0

Views

Author

Ilya Gutkovskiy, Jun 27 2024

Keywords

Crossrefs

Cf. A081360, A081362, A109697, A280454, A284314, A374064, A374065, A374076, A374077, A374078.

Programs

  • Mathematica
    nmax = 75; CoefficientList[Series[Product[1/(1 + x^(5 k - 4)), {k, 1, nmax}], {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = (1/n) Sum[DivisorSum[k, (-1)^(k/#) # &, Mod[#, 5] == 1 &] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 75}]
Showing 1-5 of 5 results.

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