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Rows converge The sequence of rows converges to A066571.

1

1 1

1 3 1

1 3 3 1

1 3 7 3 1

1 3 9 9 3 1

1 3 9 19 9 3 1

1 3 9 25 25 9 3 1

1 3 9 29 51 29 9 3 1

1 3 9 31 75 75 31 9 3 1

1 3 9 31 93 151 93 31 9 3 1

allocated for Gus WisemanTriangle read by rows where T(n,k) is the number of nonempty subsets of {1..n} with mean k.

1, 1, 1, 1, 3, 1, 1, 3, 3, 1, 1, 3, 7, 3, 1, 1, 3, 9, 9, 3, 1, 1, 3, 9, 19, 9, 3, 1, 1, 3, 9, 25, 25, 9, 3, 1, 1, 3, 9, 29, 51, 29, 9, 3, 1, 1, 3, 9, 31, 75, 75, 31, 9, 3, 1, 1, 3, 9, 31, 93, 151, 93, 31, 9, 3, 1, 1, 3, 9, 31, 105, 235, 235, 105, 31, 9, 3, 1

1,5

All terms are odd.

Triangle begins:

1

1 1

1 3 1

1 3 3 1

1 3 7 3 1

1 3 9 9 3 1

1 3 9 19 9 3 1

1 3 9 25 25 9 3 1

1 3 9 29 51 29 9 3 1

1 3 9 31 75 75 31 9 3 1

1 3 9 31 93 151 93 31 9 3 1

1 3 9 31 105 235 235 105 31 9 3 1

The subsets counted in row n = 5:

{1} {2} {3} {4} {5}

{1,3} {1,5} {3,5}

{1,2,3} {2,4} {3,4,5}

{1,3,5}

{2,3,4}

{1,2,4,5}

{1,2,3,4,5}

Table[Length[Select[Subsets[Range[n]], Mean[#]==k&]], {n, 10}, {k, n}]

Row sums are A051293.

Rows converge to A066571.

The version for partitions is A327482.

Cf. A000016, A063776, A065795, A135342, A327474, A327475, A327483, A327484.

allocated

nonn,tabl

Gus Wiseman, Sep 13 2019

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allocated for Gus Wiseman

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