A374060 Expansion of Product_{k>=1} (1 - x^(3*k-1)) * (1 - x^(3*k)).

1, 0, -1, -1, 0, 0, -1, 1, 1, 0, 0, 1, 0, 0, 1, 0, -1, 0, 0, -1, 0, 1, -1, -1, 0, -1, 0, 1, 0, -1, 1, 0, -1, 1, 1, -1, 0, 1, 0, 1, 1, -1, 0, 1, -1, -1, 2, 0, -1, 1, 0, -1, 1, 0, -2, 0, 0, -1, 1, 1, -2, 0, 1, -2, 0, 2, -1, -1, 1, -1, -1, 2, -1, -1, 2, 0, -1, 2, 1, -2, 1, 0, -2, 2, 1, -2

Offset: 0

Author

Ilya Gutkovskiy, Jun 27 2024

Keywords

• sign

Links

• Ilya Gutkovskiy, Scatterplot of a(n) up to n=10000

Crossrefs

Cf. A010815, A035361, A035382, A081362, A082050, A132462, A137569, A274719, A284315, A374058.

Programs

Mathematica

nmax = 85; CoefficientList[Series[Product[(1 - x^(3 k - 1)) (1 - x^(3 k)), {k, 1, nmax}], {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = -(1/n) Sum[Plus @@ Select[Divisors[k], Mod[#, 3] != 1 &] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 85}]

Formula

a(0) = 1; a(n) = -(1/n) * Sum_{k=1..n} A082050(k) * a(n-k).

a(0) = 1; a(n) = -Sum_{k=1..n} A035361(k) * a(n-k).

a(n) = Sum_{k=0..n} A010815(k) * A035382(n-k).