A374064 Expansion of Product_{k>=1} 1 / (1 + x^(3*k-1)).

1, 0, -1, 0, 1, -1, -1, 1, 0, -1, 1, 0, -1, 1, 0, -2, 2, 1, -3, 1, 3, -3, 0, 3, -3, -1, 4, -3, -1, 5, -3, -3, 7, -3, -5, 7, -1, -7, 8, 0, -8, 8, 1, -11, 10, 3, -14, 9, 8, -17, 8, 10, -18, 6, 14, -22, 6, 19, -24, 1, 26, -26, -3, 30, -25, -9, 37, -27, -13, 42, -26, -23, 51, -25, -31, 56

Offset: 0

Author

Ilya Gutkovskiy, Jun 27 2024

Keywords

• sign

Crossrefs

Cf. A035386, A081360, A081362, A109389, A132463, A261612, A262928, A284315, A374065.

Programs

Mathematica

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

Formula

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

a(n) = Sum_{k=0..n} A081362(k) * A132463(n-k).

a(n) = Sum_{k=0..n} A109389(k) * A261612(n-k).