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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). 2
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 (list; graph; refs; listen; history; text; internal format)
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 A244208 A296329

Adjacent sequences:  A284939 A284940 A284941 * A284943 A284944 A284945

KEYWORD

nonn

AUTHOR

Ilya Gutkovskiy, Apr 06 2017

STATUS

approved

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Last modified August 10 12:58 EDT 2022. Contains 356039 sequences. (Running on oeis4.)