A374077 -id:A374077 - cp's OEIS frontend

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

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

Ilya Gutkovskiy, Jun 27 2024

sign

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

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}]

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

Ilya Gutkovskiy, Jun 27 2024

sign

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

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

Ilya Gutkovskiy, Jun 27 2024

sign

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

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}]

cp's OEIS Frontend

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

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

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Mathematica

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

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Mathematica