A322291 Triangle T read by rows: T(n, k) = Sum_{i=1..k} binomial(n, floor((n-k)/2)+i).
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, 119, 126, 127, 70, 126, 182, 210, 238, 246, 254, 255, 126, 252, 336, 420, 456, 492, 501, 510, 511, 252, 462, 672, 792, 912, 957, 1002, 1012, 1022, 1023, 462, 924, 1254, 1584, 1749, 1914, 1969, 2024, 2035, 2046, 2047
Offset: 1
Examples
n\k| 1 2 3 4 5 6 ---+----------------------------- 1 | 1 2 | 2 3 3 | 3 6 7 4 | 6 10 14 15 5 | 10 20 25 30 31 6 | 20 35 50 56 62 63 ...
Links
- John Tyler Rascoe, Rows n = 1..141 of the triangle, flattened
- P. Erdős, On a lemma of Littlewood and Offord, Bull. Amer. Math. Soc., 51 (1945), 898-902.
- C. Pelekis and V. Vlasák, On k-antichains in the unit n-cube, arXiv:1908.04727 [math.CA], 2019.
Crossrefs
Programs
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GAP
Flat(List([1..11], n->List([1..n], k->Sum([1..k], i->Binomial(n, Int((n-k)/2)+i)))));
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Maple
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);
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Mathematica
T[n_,k_]:=Sum[Binomial[n,Floor[(n-k)/2]+i],{i,1,k}]; Table[T[n,k],{n,1,11},{k,1,n}] -
PARI
T(n, k) = sum(i=1, k, binomial(n, floor((n-k)/2)+i));
Comments