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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

%I #37 Jun 27 2022 23:34:59

%S 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,

%T 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,

%U 54,123,156,111,41,6,1,14,64,161,227,180,74,12

%N 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).

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

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

%H Marcel K. Goh and Jonah Saks, <a href="https://arxiv.org/abs/2206.12535">Alternating-sum statistics for certain sets of integers</a>, arXiv:2206.12535 [math.CO], 2022.

%F Sum_{k=1..ceiling(n/2)} k * T(n,k) = A087077(n). - _Alois P. Heinz_, Jun 24 2022

%e Triangle T(n,k) begins:

%e n/k 0 1 2 3 4 5 6 7 8 9 10 11 12

%e 0 1

%e 1 1 1

%e 2 1 2

%e 3 1 3 1

%e 4 1 4 2

%e 5 1 5 5 2

%e 6 1 6 7 3

%e 7 1 7 12 10 3

%e 8 1 8 16 15 5

%e 9 1 9 22 26 13 2

%e 10 1 10 28 38 22 4

%e 11 1 11 37 66 60 26 4

%e 12 1 12 43 80 76 35 6

%e ...

%e For n=6 and k=3 the T(6,3) = 3 primitive sets are {2,3,5}, {3,4,5}, and {4,5,6}.

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

%Y Row sums give A051026.

%Y T(2n,n) gives A174094.

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

%Y Cf. A087077, A087086.

%K nonn,tabf

%O 0,5

%A _Marcel K. Goh_, Jun 20 2022

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Last modified March 29 02:23 EDT 2024. Contains 371264 sequences. (Running on oeis4.)