A316231 Expansion of Product_{k>=1} 1/(1 + q(k)*x^k), where q(k) = number of partitions of k into distinct parts (A000009).

1, -1, 0, -2, 1, -2, 3, -3, 6, -8, 14, -10, 28, -26, 41, -73, 90, -112, 155, -221, 288, -501, 560, -799, 1153, -1610, 1953, -3095, 4073, -5224, 7295, -9536, 13536, -18402, 24757, -32936, 48714, -60790, 82101, -113247, 153330, -201522, 275713, -367041, 492991

Offset: 0

Author

Ilya Gutkovskiy, Jun 27 2018

Keywords

• sign

Links

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

Crossrefs

Cf. A000009, A089254, A270995, A279785, A304786, A316230.

Programs

Mathematica

nmax = 44; CoefficientList[Series[Product[1/(1 + PartitionsQ[k] x^k), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 44; CoefficientList[Series[Exp[Sum[Sum[(-1)^k PartitionsQ[j]^k x^(j k)/k, {j, 1, nmax}], {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, 44}]

Formula

G.f.: exp(Sum_{k>=1} Sum_{j>=1} (-1)^k*q(j)^k*x^(j*k)/k).