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%I #35 Jun 04 2023 08:40:56
%S 1,2,3,3,6,7,6,10,14,15,10,20,25,30,31,20,35,50,56,62,63,35,70,91,112,
%T 119,126,127,70,126,182,210,238,246,254,255,126,252,336,420,456,492,
%U 501,510,511,252,462,672,792,912,957,1002,1012,1022,1023,462,924,1254,1584,1749,1914,1969,2024,2035,2046,2047
%N Triangle T read by rows: T(n, k) = Sum_{i=1..k} binomial(n, floor((n-k)/2)+i).
%C T(n, k) is a sharp upper bound on the cardinality of a k-antichain in {0, 1}^n due to P. Erdős.
%C T(n, k) is also the total number of compositions with first part k, n+1 parts, and all differences between adjacent parts in {-1,1}. - _John Tyler Rascoe_, May 07 2023
%H John Tyler Rascoe, <a href="/A322291/b322291.txt">Rows n = 1..141 of the triangle, flattened</a>
%H P. Erdős, <a href="https://doi.org/10.1090/S0002-9904-1945-08454-7">On a lemma of Littlewood and Offord</a>, Bull. Amer. Math. Soc., 51 (1945), 898-902.
%H C. Pelekis and V. Vlasák, <a href="https://arxiv.org/abs/1908.04727">On k-antichains in the unit n-cube</a>, arXiv:1908.04727 [math.CA], 2019.
%F T(n, n) = A000225(n).
%F T(n, n-1) = A000918(n).
%F T(n, n-2) = A000247(n).
%F T(n, n-3) = A052515(n).
%F T(n, n-4) = A272352(n+1).
%F T(n, n-5) = A052516(n).
%e n\k| 1 2 3 4 5 6
%e ---+-----------------------------
%e 1 | 1
%e 2 | 2 3
%e 3 | 3 6 7
%e 4 | 6 10 14 15
%e 5 | 10 20 25 30 31
%e 6 | 20 35 50 56 62 63
%e ...
%p a:=(n, k)->sum(binomial(n, floor((1/2)*n-(1/2)*k)+i), i = 1..k): seq(seq(a(n, k), k = 1..n), n = 1..11);
%t T[n_,k_]:=Sum[Binomial[n,Floor[(n-k)/2]+i],{i,1,k}]; Table[T[n,k],{n,1,11},{k,1,n}]
%o (GAP) Flat(List([1..11], n->List([1..n], k->Sum([1..k], i->Binomial(n, Int((n-k)/2)+i)))));
%o (PARI) T(n, k) = sum(i=1, k, binomial(n, floor((n-k)/2)+i));
%Y Cf. A000225 (diagonal), A189390 (row sums).
%Y Cf. A000247, A000918, A001405, A006126, A007318, A052515, A052516, A263857, A272352, A306550.
%K nonn,tabl
%O 1,2
%A _Stefano Spezia_, Aug 28 2019