A304783 Expansion of Product_{k>=1} (1 - x^k)^q(k), where q(k) = number of partitions of k into distinct parts (A000009).

1, -1, -1, -1, 0, 1, 0, 3, 2, 3, 1, 3, -2, 0, -6, -8, -12, -14, -18, -19, -19, -15, -3, 4, 29, 46, 90, 114, 165, 192, 248, 252, 276, 232, 185, 29, -143, -454, -811, -1324, -1909, -2609, -3348, -4132, -4851, -5386, -5653, -5380, -4470, -2477, 664, 5582, 12193, 21314

Offset: 0

Author

Ilya Gutkovskiy, May 18 2018

Keywords

• sign

Comments

Convolution inverse of A089259.

Links

• Eric Weisstein's World of Mathematics, Partition Function Q
• Index entries for sequences related to partitions

Crossrefs

Cf. A000009, A050342, A089254, A089259, A300508.

Programs

Mathematica

nmax = 53; CoefficientList[Series[Product[(1 - x^k)^PartitionsQ[k], {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[-Sum[d PartitionsQ[d], {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 53}]

Formula

G.f.: Product_{k>=1} (1 - x^k)^A000009(k).