A284319 -id:A284319 - cp's OEIS frontend

A284320 Expansion of Product_{k>=0} (1 - x^(5*k+3)) in powers of x.

1, 0, 0, -1, 0, 0, 0, 0, -1, 0, 0, 1, 0, -1, 0, 0, 1, 0, -1, 0, 0, 2, 0, -1, -1, 0, 2, 0, -1, -1, 0, 3, 0, -1, -2, 0, 3, 0, -1, -3, 0, 4, 1, -1, -4, 0, 4, 1, -1, -5, 0, 5, 2, -1, -7, 0, 5, 3, -1, -8, 0, 6, 5, -1, -10, -1, 6, 6, -1, -12, -1, 7, 9, -1, -14, -2, 7, 11

Offset: 0

Seiichi Manyama, Mar 25 2017

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G. C. Greubel, Table of n, a(n) for n = 0..10000

Cf. Product_{k>=0} (1 - x^(5*k+m)): A284314 (m=1), A284319 (m=2), this sequence (m=3), A284317 (m=4).

Cf. A281271, A284281.

CoefficientList[Series[Product[1 - x^(5k + 3), {k, 0, 100}], {x, 0, 100}], x] (* Indranil Ghosh, Mar 25 2017 *) (* or *)
a[0]=1; a[n_]:=a[n]= -(1/n) Sum[ a[n-k] DivisorSum[k, # &, Mod[#,5] == 3 &], {k, n}]; a /@ Range[0, 100] (* Giovanni Resta, Mar 25 2017 *)

Vec(prod(k=0, 100, 1 - x^(5*k + 3)) + O(x^101)) \\ Indranil Ghosh, Mar 25 2017

a(n) = -(1/n)*Sum_{k=1..n} A284281(k)*a(n-k), a(0) = 1.

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

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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}]

A284322 Expansion of Product_{k>=0} (1 - x^(5k+2))(1 - x^(5*k+3)) in powers of x.

Original entry on oeis.org

1, 0, -1, -1, 0, 1, 0, -1, -1, 1, 2, 1, -2, -2, 1, 3, 1, -3, -3, 2, 5, 1, -5, -5, 2, 7, 2, -7, -7, 4, 11, 3, -11, -11, 5, 15, 4, -15, -14, 8, 22, 6, -21, -21, 10, 30, 8, -29, -28, 15, 42, 11, -40, -39, 19, 56, 15, -53, -51, 27, 76, 20, -72, -70, 34, 99, 26, -94, -90

Offset: 0

Seiichi Manyama, Mar 25 2017

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Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A003106, A284152, A284319, A284320, A284321.

CoefficientList[Series[Product[(1 - x^(5k + 2)) ( 1 - x^(5k + 3)), {k, 0, 100}], {x, 0, 100}],x] (* Indranil Ghosh, Mar 25 2017 *)

Vec(prod(k=0, 100, (1 - x^(5*k + 2)) * (1 - x^(5*k + 3))) + O(x^101)) \\ Indranil Ghosh, Mar 25 2017

a(n) = -(1/n)*Sum_{k=1..n} A284152(k)*a(n-k), a(0) = 1.

cp's OEIS Frontend

A284320 Expansion of Product_{k>=0} (1 - x^(5*k+3)) in powers of x.

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

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A284322 Expansion of Product_{k>=0} (1 - x^(5k+2))(1 - x^(5*k+3)) in powers of x.

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cp's OEIS Frontend

A284320 Expansion of Product_{k>=0} (1 - x^(5*k+3)) in powers of x.

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

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A284322 Expansion of Product_{k>=0} (1 - x^(5*k+2))*(1 - x^(5*k+3)) in powers of x.

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A284322 Expansion of Product_{k>=0} (1 - x^(5k+2))(1 - x^(5*k+3)) in powers of x.