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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).
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
0,5
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.
LINKS
Marcel K. Goh and Jonah Saks, Alternating-sum statistics for certain sets of integers, arXiv:2206.12535 [math.CO], 2022.
FORMULA
Sum_{k=1..ceiling(n/2)} k * T(n,k) = A087077(n). - Alois P. Heinz, Jun 24 2022
EXAMPLE
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}.
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.
Sequence in context: A174066 A089178 A187489 * A116599 A138121 A138151
KEYWORD
nonn,tabf
AUTHOR
Marcel K. Goh, Jun 20 2022
STATUS
approved