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%I #31 Dec 10 2023 17:37:46
%S 1,1,1,3,1,1,4,4,1,1,10,5,5,1,1,15,15,6,6,1,1,35,21,21,7,7,1,1,56,56,
%T 28,28,8,8,1,1,126,84,84,36,36,9,9,1,1,210,210,120,120,45,45,10,10,1,
%U 1,462,330,330,165,165,55,55,11,11,1,1,792,792,495,495,220,220,66,66,12,12,1,1
%N Triangle read by rows: T(n, k) = binomial(n, floor((n+k+1)/2)) with k <= n.
%C T(n, k) is a tight upper bound of the cardinality of an intersecting Sperner family or antichain of the set {1, 2,..., n}, where every collection of pairwise independent subsets is characterized by an intersection of cardinality at least k (see Theorem 1.3 in Wong and Tay).
%C Equals A061554 with the first row of the array (resp. the first column of the triangle) removed. - _Georg Fischer_, Jul 26 2023
%H Eric Charles Milner, <a href="https://doi.org/10.1112/jlms/s1-43.1.204">A Combinatorial Theorem On Systems of Sets</a>, Journal of the London Mathematical Society, 43, (1968), 204-206.
%H W. H. W. Wong and E. G. Tay, <a href="https://arxiv.org/abs/2001.01910">On Cross-intersecting Sperner Families</a>, arXiv:2001.01910 [math.CO], 2020.
%H <a href="/index/Am#antichains">Index entries for sequences related to antichains</a>.
%F T(n, k) = A007318(n, A004526(n+k+1)) with k <= n.
%e The triangle T(n, k) begins
%e n\k| 1 2 3 4 5 6 7 8
%e ---+-------------------------------
%e 1 | 1
%e 2 | 1 1
%e 3 | 3 1 1
%e 4 | 4 4 1 1
%e 5 | 10 5 5 1 1
%e 6 | 15 15 6 6 1 1
%e 7 | 35 21 21 7 7 1 1
%e 8 | 56 56 28 28 8 8 1 1
%e ...
%t T[n_,k_]:=Binomial[n,Floor[(n+k+1)/2]]; Table[T[n,k],{n,12},{k,n}]//Flatten
%o (PARI) T(n, k) = binomial(n, (n+k+1)\2);
%o vector(10, n, vector(n, k, T(n, k))) \\ _Michel Marcus_, Jun 01 2020
%Y Cf. A000372, A001405, A004526, A007318, A007695, A061554, A266696, A325982, A325983.
%Y Cf. A037951 (k=3), A037952 (k=1), A037953 (k=5), A037954 (k=7), A037955 (k=2), A037956 (k=4), A037957 (k=6), A037958 (k=8), A045621 (row sums).
%K nonn,tabl
%O 1,4
%A _Stefano Spezia_, May 31 2020
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