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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).

%I A355145 #37 Jun 27 2022 23:34:59
%S A355145 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 A355145 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 A355145 54,123,156,111,41,6,1,14,64,161,227,180,74,12
%N 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).
%C A355145 A set is primitive if it does not contain distinct i and j such that i divides j.
%C A355145 For n >= 2, the alternating row sums equal -1.
%H A355145 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 A355145 Sum_{k=1..ceiling(n/2)} k * T(n,k) = A087077(n). - _Alois P. Heinz_, Jun 24 2022
%e A355145 Triangle T(n,k) begins:
%e A355145    n/k 0  1  2  3  4  5  6  7  8  9 10 11 12
%e A355145     0  1
%e A355145     1  1  1
%e A355145     2  1  2
%e A355145     3  1  3  1
%e A355145     4  1  4  2
%e A355145     5  1  5  5  2
%e A355145     6  1  6  7  3
%e A355145     7  1  7 12 10  3
%e A355145     8  1  8 16 15  5
%e A355145     9  1  9 22 26 13  2
%e A355145    10  1 10 28 38 22  4
%e A355145    11  1 11 37 66 60 26  4
%e A355145    12  1 12 43 80 76 35  6
%e A355145    ...
%e A355145 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 A355145 Columns k=0..2 give: A000012, A000027, A161664.
%Y A355145 Row sums give A051026.
%Y A355145 T(2n,n) gives A174094.
%Y A355145 T(2n-1,n) gives A192298 for n>=1.
%Y A355145 Cf. A087077, A087086.
%K A355145 nonn,tabf
%O A355145 0,5
%A A355145 _Marcel K. Goh_, Jun 20 2022