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A355147
Triangle read by rows: T(n,k) is the number of product-free subsets of {1,...,n} with cardinality k; n >= 0, 0 <= k <= A028391(n).
0
1, 1, 1, 1, 1, 2, 1, 1, 3, 2, 1, 4, 5, 2, 1, 5, 9, 6, 1, 1, 6, 14, 15, 7, 1, 1, 7, 20, 29, 22, 8, 1, 1, 8, 26, 43, 38, 17, 3, 1, 9, 34, 68, 76, 47, 15, 2, 1, 10, 43, 102, 144, 123, 62, 17, 2, 1, 11, 53, 143, 234, 238, 149, 55, 11, 1, 1, 12, 64, 196, 377, 472, 387, 204, 66, 12, 1
OFFSET
0,6
COMMENTS
S is product-free if for any i,j in S, not necessarily distinct, i*j is not in S.
For n >= 2, the alternating row sums give 0.
LINKS
Marcel K. Goh and Jonah Saks, Alternating-sum statistics for certain sets of integers, arXiv:2206.12535 [math.CO], 2022.
EXAMPLE
Triangle T(n,k) begins:
n/k 0 1 2 3 4 5 6 7 8 9
0 1
1 1
2 1 1
3 1 2 1
4 1 3 2
5 1 4 5 2
6 1 5 9 6 1
7 1 6 14 15 7 1
8 1 7 20 29 22 8 1
9 1 8 26 43 38 17 3
10 1 9 34 68 76 47 15 2
11 1 10 43 102 144 123 62 17 2
12 1 11 53 143 234 238 149 55 11 1
...
For n=5 and k=3 the T(5,3) = 2 sets are {2,3,5} and {3,4,5}.
CROSSREFS
Row sums give A326489.
Cf. A028391.
Sequence in context: A348445 A023996 A307998 * A049998 A029253 A288165
KEYWORD
nonn,tabf
AUTHOR
Marcel K. Goh, Jun 28 2022
STATUS
approved