A292561 Expansion of Product_{k>=1} (1 - mu(k)^2*x^k), where mu() is the Moebius function (A008683).

1, -1, -1, 0, 1, 0, -1, 1, 2, 0, -3, 0, 2, 0, -3, 0, 5, 0, -4, -2, 4, 0, -5, 0, 7, 3, -8, -1, 5, 1, -10, 0, 13, 2, -10, -3, 14, -2, -17, -3, 21, 5, -22, 0, 22, 4, -34, -5, 33, 9, -33, -10, 43, 6, -43, -19, 52, 16, -51, -13, 56, 24, -71, -20, 64, 26, -78, -24, 90, 24, -90, -39, 112, 26, -115, -37

Offset: 0

Author

Ilya Gutkovskiy, Sep 19 2017

Keywords

• sign
• look

Comments

Convolution inverse of A073576.

The difference between the number of partitions of n into an even number of distinct squarefree parts and the number of partitions of n into an odd number of distinct squarefree parts.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..10000
• Index entries for related partition-counting sequences

Crossrefs

Cf. A005117, A008683, A046675, A073576, A087188.

Programs

Maple

with(numtheory):
b:= proc(n) option remember; `if`(n=0, 1, add(add(d*
      abs(mobius(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..80);  # Alois P. Heinz, Sep 20 2017

Mathematica

nmax = 75; CoefficientList[Series[Product[1 - MoebiusMu[k]^2 x^k, {k, 1, nmax}], {x, 0, nmax}], x]

Formula

G.f.: Product_{k>=1} (1 - x^A005117(k)).