A300275 G.f.: 1 + Sum_{n>=1} a(n)*x^n/(1 - x^n) = Product_{n>=1} 1/(1 - x^n)^n.

1, 2, 5, 10, 23, 40, 85, 147, 276, 474, 858, 1421, 2484, 4079, 6850, 11137, 18333, 29277, 47329, 74768, 118703, 185614, 290782, 449568, 696009, 1066258, 1632376, 2479057, 3759611, 5661568, 8512308, 12722132, 18974109, 28157619, 41690937, 61453929, 90379783

Offset: 1

Author

Ilya Gutkovskiy, Mar 01 2018

Keywords

• nonn

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

Crossrefs

Cf. A000219, A000837, A078374, A300274, A300276, A300277, A300278.

Cf. A100953, A303546, A320802, A323584, A323585, A323587.

Programs

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}]

Formula

a(n) = Sum_{d|n} mu(n/d)*A000219(d).