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

%I A300275 #13 Jan 22 2019 07:45:32
%S A300275 1,2,5,10,23,40,85,147,276,474,858,1421,2484,4079,6850,11137,18333,
%T A300275 29277,47329,74768,118703,185614,290782,449568,696009,1066258,1632376,
%U A300275 2479057,3759611,5661568,8512308,12722132,18974109,28157619,41690937,61453929,90379783
%N A300275 G.f.: 1 + Sum_{n>=1} a(n)*x^n/(1 - x^n) = Product_{n>=1} 1/(1 - x^n)^n.
%C A300275 Moebius transform of A000219.
%C A300275 From _Gus Wiseman_, Jan 21 2019: (Start)
%C A300275 Also the number of plane partitions of n with relatively prime entries. For example, the a(4) = 10 plane partitions are:
%C A300275   31   211   1111
%C A300275 .
%C A300275   3   21   11   111
%C A300275   1   1    11   1
%C A300275 .
%C A300275   2   11
%C A300275   1   1
%C A300275   1   1
%C A300275 .
%C A300275   1
%C A300275   1
%C A300275   1
%C A300275   1
%C A300275 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:
%C A300275   4   31   22   211   1111
%C A300275 .
%C A300275   3   21   111
%C A300275   1   1    1
%C A300275 .
%C A300275   2   11
%C A300275   1   1
%C A300275   1   1
%C A300275 (End)
%H A300275 Alois P. Heinz, <a href="/A300275/b300275.txt">Table of n, a(n) for n = 1..10000</a>
%H A300275 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F A300275 a(n) = Sum_{d|n} mu(n/d)*A000219(d).
%p A300275 with(numtheory):
%p A300275 b:= proc(n) option remember; `if`(n=0, 1, add(
%p A300275       b(n-j)*sigma[2](j), j=1..n)/n)
%p A300275     end:
%p A300275 a:= n-> add(b(d)*mobius(n/d), d=divisors(n)):
%p A300275 seq(a(n), n=1..40);  # _Alois P. Heinz_, Jun 21 2018
%t A300275 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
%t A300275 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}]
%Y A300275 Cf. A000219, A000837, A078374, A300274, A300276, A300277, A300278.
%Y A300275 Cf. A100953, A303546, A320802, A323584, A323585, A323587.
%K A300275 nonn
%O A300275 1,2
%A A300275 _Ilya Gutkovskiy_, Mar 01 2018