A304786 Expansion of Product_{k>=1} (1 - q(k)*x^k), where q(k) = number of partitions of k into distinct parts (A000009).

1, -1, -1, -1, 0, 1, -1, 4, 2, 3, 1, 8, -8, 10, -8, -9, -15, -6, -46, -14, -65, -28, 14, -29, -43, -37, 298, 59, 234, 165, 738, 354, 1083, 703, 1944, -2024, 1917, -1085, 3658, -2385, -6421, -7220, 118, -15569, -11604, -19162, -9448, -36140, -24561, -50505, -24807, 47645

Offset: 0

Author

Ilya Gutkovskiy, May 18 2018

Keywords

• sign

Comments

Convolution inverse of A270995.

Links

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

Crossrefs

Cf. A000009, A270995, A271619, A304783, A304785.

Programs

Mathematica

nmax = 51; CoefficientList[Series[Product[(1 - PartitionsQ[k] x^k), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[-Sum[d PartitionsQ[d]^(k/d), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 51}]

Formula

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