cp's OEIS Frontend

A335322 Triangle read by rows: T(n, k) = binomial(n, floor((n+k+1)/2)) with k <= n.

%I A335322 #31 Dec 10 2023 17:37:46
%S A335322 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 A335322 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 A335322 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 A335322 Triangle read by rows: T(n, k) = binomial(n, floor((n+k+1)/2)) with k <= n.
%C A335322 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 A335322 Equals A061554 with the first row of the array (resp. the first column of the triangle) removed. - _Georg Fischer_, Jul 26 2023
%H A335322 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 A335322 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 A335322 <a href="/index/Am#antichains">Index entries for sequences related to antichains</a>.
%F A335322 T(n, k) = A007318(n, A004526(n+k+1)) with k <= n.
%e A335322 The triangle T(n, k) begins
%e A335322 n\k|  1   2   3   4   5   6   7   8
%e A335322 ---+-------------------------------
%e A335322 1  |  1
%e A335322 2  |  1   1
%e A335322 3  |  3   1   1
%e A335322 4  |  4   4   1   1
%e A335322 5  | 10   5   5   1   1
%e A335322 6  | 15  15   6   6   1   1
%e A335322 7  | 35  21  21   7   7   1   1
%e A335322 8  | 56  56  28  28   8   8   1   1
%e A335322 ...
%t A335322 T[n_,k_]:=Binomial[n,Floor[(n+k+1)/2]]; Table[T[n,k],{n,12},{k,n}]//Flatten
%o A335322 (PARI) T(n, k) = binomial(n, (n+k+1)\2);
%o A335322 vector(10, n, vector(n, k, T(n, k))) \\ _Michel Marcus_, Jun 01 2020
%Y A335322 Cf. A000372, A001405, A004526, A007318, A007695, A061554, A266696, A325982, A325983.
%Y A335322 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 A335322 nonn,tabl
%O A335322 1,4
%A A335322 _Stefano Spezia_, May 31 2020