A355145 Triangle read by rows: T(n,k) is the number of primitive subsets of {1,...,n} of cardinality k; n>=0, 0<=k<=ceiling(n/2).

1, 1, 1, 1, 2, 1, 3, 1, 1, 4, 2, 1, 5, 5, 2, 1, 6, 7, 3, 1, 7, 12, 10, 3, 1, 8, 16, 15, 5, 1, 9, 22, 26, 13, 2, 1, 10, 28, 38, 22, 4, 1, 11, 37, 66, 60, 26, 4, 1, 12, 43, 80, 76, 35, 6, 1, 13, 54, 123, 156, 111, 41, 6, 1, 14, 64, 161, 227, 180, 74, 12

Offset: 0

Author

Marcel K. Goh, Jun 20 2022

Keywords

• nonn
• tabf

Comments

A set is primitive if it does not contain distinct i and j such that i divides j.

For n >= 2, the alternating row sums equal -1.

Examples

			Triangle T(n,k) begins:
   n/k 0  1  2  3  4  5  6  7  8  9 10 11 12
    0  1
    1  1  1
    2  1  2
    3  1  3  1
    4  1  4  2
    5  1  5  5  2
    6  1  6  7  3
    7  1  7 12 10  3
    8  1  8 16 15  5
    9  1  9 22 26 13  2
   10  1 10 28 38 22  4
   11  1 11 37 66 60 26  4
   12  1 12 43 80 76 35  6
   ...
For n=6 and k=3 the T(6,3) = 3 primitive sets are {2,3,5}, {3,4,5}, and {4,5,6}.
		

Links

• Marcel K. Goh and Jonah Saks, Alternating-sum statistics for certain sets of integers, arXiv:2206.12535 [math.CO], 2022.

Crossrefs

Columns k=0..2 give: A000012, A000027, A161664.

Row sums give A051026.

T(2n,n) gives A174094.

T(2n-1,n) gives A192298 for n>=1.

Cf. A087077, A087086.

Formula

Sum_{k=1..ceiling(n/2)} k * T(n,k) = A087077(n). - Alois P. Heinz, Jun 24 2022