%I #17 Aug 07 2019 17:28:20

%S 1,3,8,19,46,107,244,547,1213,2665,5807,12567,27042,57899,123428,

%T 262115,554750,1170538,2463154,5170462,10829234,22635087,47223412,

%U 98353299,204519549,424665001,880581806,1823667221,3772341661,7794697759,16089424392,33178906531,68357928558

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

%C Total number of squarefree parts in all compositions (ordered partitions) of n.

%H Alois P. Heinz, <a href="/A284942/b284942.txt">Table of n, a(n) for n = 1..3312</a>

%H <a href="/index/Com#comp">Index entries for sequences related to compositions</a>

%F G.f.: Sum_{k>=1} mu(k)^2*x^k*(1 - x)^2/(1 - 2*x)^2.

%e a(4) = 19 because we have [4], [3, 1], [2, 2], [2, 1, 1], [1, 3], [1, 2, 1], [1, 1, 2], [1, 1, 1, 1] and 0 + 2 + 2 + 3 + 2 + 3 + 3 + 4 = 19.

%p a:= proc(n) option remember; add(`if`(numtheory[

%p issqrfree](j), ceil(2^(n-j-1)), 0)+a(n-j), j=1..n)

%p end:

%p seq(a(n), n=1..33); # _Alois P. Heinz_, Aug 07 2019

%t nmax = 33; Rest[CoefficientList[Series[Sum[MoebiusMu[k]^2 x^k (1 - x)^2/(1 - 2 x)^2, {k, 1, nmax}], {x, 0, nmax}], x]]

%o (PARI) x='x+O('x^34); Vec(sum(k=1, 34, moebius(k) ^2*x^k*(1 - x)^2/(1 - 2*x)^2)) \\ _Indranil Ghosh_, Apr 06 2017

%Y Cf. A005117, A008683, A011782, A059570, A097941, A097979, A102291, A281573, A284943.

%K nonn

%O 1,2

%A _Ilya Gutkovskiy_, Apr 06 2017