A284942 Expansion of Sum_{k>=1} mu(k)^2*x^k*(1 - x)^2/(1 - 2*x)^2, where mu() is the Moebius function (A008683).

1, 3, 8, 19, 46, 107, 244, 547, 1213, 2665, 5807, 12567, 27042, 57899, 123428, 262115, 554750, 1170538, 2463154, 5170462, 10829234, 22635087, 47223412, 98353299, 204519549, 424665001, 880581806, 1823667221, 3772341661, 7794697759, 16089424392, 33178906531, 68357928558

Offset: 1

Author

Ilya Gutkovskiy, Apr 06 2017

Keywords

• nonn

Comments

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

Examples

			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.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..3312
• Index entries for sequences related to compositions

Crossrefs

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

Programs

Maple

a:= proc(n) option remember; add(`if`(numtheory[
      issqrfree](j), ceil(2^(n-j-1)), 0)+a(n-j), j=1..n)
    end:
seq(a(n), n=1..33);  # Alois P. Heinz, Aug 07 2019

Mathematica

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]]

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

Formula

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