OFFSET
1,2
COMMENTS
Moebius transform of A000219.
From Gus Wiseman, Jan 21 2019: (Start)
Also the number of plane partitions of n with relatively prime entries. For example, the a(4) = 10 plane partitions are:
31 211 1111
.
3 21 11 111
1 1 11 1
.
2 11
1 1
1 1
.
1
1
1
1
Also the number of plane partitions of n whose multiset of rows is aperiodic, meaning its multiplicities are relatively prime. For example, the a(4) = 10 plane partitions are:
4 31 22 211 1111
.
3 21 111
1 1 1
.
2 11
1 1
1 1
(End)
LINKS
Alois P. Heinz, Table of n, a(n) for n = 1..10000
N. J. A. Sloane, Transforms
FORMULA
a(n) = Sum_{d|n} mu(n/d)*A000219(d).
MAPLE
with(numtheory):
b:= proc(n) option remember; `if`(n=0, 1, add(
b(n-j)*sigma[2](j), j=1..n)/n)
end:
a:= n-> add(b(d)*mobius(n/d), d=divisors(n)):
seq(a(n), n=1..40); # Alois P. Heinz, Jun 21 2018
MATHEMATICA
nn = 37; f[x_] := 1 + Sum[a[n] x^n/(1 - x^n), {n, 1, nn}]; sol = SolveAlways[0 == Series[f[x] - Product[1/(1 - x^n)^n, {n, 1, nn}], {x, 0, nn}], x]; Table[a[n], {n, 1, nn}] /. sol // Flatten
s[n_] := SeriesCoefficient[Product[1/(1 - x^k)^k, {k, 1, n}], {x, 0, n}]; a[n_] := Sum[MoebiusMu[n/d] s[d], {d, Divisors[n]}]; Table[a[n], {n, 1, 37}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Mar 01 2018
STATUS
approved