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

1,2

Comments

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

Links

Alois P. Heinz, Table of n, a(n) for n = 1..3312

Index entries for sequences related to compositions

Formula

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

Example

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.

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]]

Prog

(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

Crossrefs

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

Sequence in context: A096576 A309537 A126874 * A121811 A370033 A244208

Adjacent sequences: A284939 A284940 A284941 * A284943 A284944 A284945

Keyword

nonn

Author

Ilya Gutkovskiy, Apr 06 2017

Status

approved