%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
|