A357521 -id:A357521 - cp's OEIS frontend

A357524 Expansion of Product_{k>=1} 1 / (1 + mu(k)*x^k).

1, -1, 2, -1, 2, 0, 1, 2, 0, 3, 0, 4, 1, 4, 2, 4, 4, 4, 5, 6, 6, 6, 8, 8, 10, 9, 11, 12, 13, 14, 17, 17, 20, 19, 23, 24, 28, 27, 30, 34, 34, 40, 41, 47, 48, 50, 56, 62, 64, 71, 72, 80, 85, 91, 99, 104, 113, 112, 128, 135, 147, 153, 159, 176, 180, 196, 210, 220, 233, 240, 264

Offset: 0

Ilya Gutkovskiy, Oct 02 2022

sign

Cf. A008683, A117211, A185694, A300663, A306327, A329069, A357521, A357525.

nmax = 70; CoefficientList[Series[Product[1/(1 + MoebiusMu[k] x^k), {k, 1, nmax}], {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = (1/n) Sum[Sum[d (-MoebiusMu[d])^(k/d), {d, Divisors[k]}] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 70}]

A357525 Expansion of Product_{k>=1} (1 + mu(k)*x^k).

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Oct 02 2022

sign

Cf. A008683, A087188, A117210, A185694, A300663, A306327, A357521, A357524.

nmax = 75; CoefficientList[Series[Product[(1 + MoebiusMu[k] x^k), {k, 1, nmax}], {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = -(1/n) Sum[Sum[d (-MoebiusMu[d])^(k/d), {d, Divisors[k]}] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 75}]

cp's OEIS Frontend

A357524 Expansion of Product_{k>=1} 1 / (1 + mu(k)*x^k).

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