A304785 Expansion of Product_{k>=1} (1 - p(k)*x^k), where p(k) = number of partitions of k (A000041).

1, -1, -2, -1, -2, 4, 0, 15, 7, 17, 22, 26, -79, -2, -12, -392, -250, -392, -443, -640, -404, -795, 5106, 1147, 3304, 4542, 32330, 21001, 23372, 21015, 14496, 16165, -17213, 51296, -231330, -890169, -492310, -755449, -1648273, 131600, -6308274, -2160440, -4410945, 1593319

Offset: 0

Author

Ilya Gutkovskiy, May 18 2018

Keywords

• sign

Comments

Convolution inverse of A063834.

Links

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

Crossrefs

Cf. A000041, A063834, A271619, A300508.

Programs

Mathematica

nmax = 43; CoefficientList[Series[Product[(1 - PartitionsP[k] x^k), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[-Sum[d PartitionsP[d]^(k/d), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 43}]

Formula

G.f.: Product_{k>=1} (1 - A000041(k)*x^k).