A300508 Expansion of Product_{k>=1} (1 - x^k)^p(k), where p(k) = number of partitions of k (A000041).

1, -1, -2, -1, -1, 3, 3, 9, 9, 10, 8, -1, -21, -45, -77, -130, -163, -198, -179, -108, 101, 451, 1058, 1878, 2999, 4276, 5595, 6511, 6446, 4443, -838, -11069, -28373, -54652, -91948, -140370, -198501, -259706, -311997, -332003, -285486, -118600, 239086, 881998, 1918851, 3470261

Offset: 0

Author

Ilya Gutkovskiy, Mar 07 2018

Keywords

• sign

Comments

Convolution inverse of A001970.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..3000

Crossrefs

Cf. A000041, A001970, A261049.

Programs

Maple

with(numtheory): with(combinat):
b:= proc(n) option remember; `if`(n=0, 1, add(add(d*
      numbpart(d), d=divisors(j))*b(n-j), j=1..n)/n)
    end:
a:= proc(n) option remember; `if`(n=0, 1,
      -add(b(n-i)*a(i), i=0..n-1))
    end:
seq(a(n), n=0..60);  # Alois P. Heinz, Mar 07 2018

Mathematica

nmax = 45; CoefficientList[Series[Product[(1 - x^k)^PartitionsP[k], {k, 1, nmax}], {x, 0, nmax}], x]

Formula

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