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A322291
Triangle T read by rows: T(n, k) = Sum_{i=1..k} binomial(n, floor((n-k)/2)+i).
2
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
1,2
COMMENTS
T(n, k) is a sharp upper bound on the cardinality of a k-antichain in {0, 1}^n due to P. Erdős.
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
LINKS
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.
FORMULA
T(n, n) = A000225(n).
T(n, n-1) = A000918(n).
T(n, n-2) = A000247(n).
T(n, n-3) = A052515(n).
T(n, n-4) = A272352(n+1).
T(n, n-5) = A052516(n).
EXAMPLE
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
...
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);
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}]
PROG
(GAP) Flat(List([1..11], n->List([1..n], k->Sum([1..k], i->Binomial(n, Int((n-k)/2)+i)))));
(PARI) T(n, k) = sum(i=1, k, binomial(n, floor((n-k)/2)+i));
CROSSREFS
Cf. A000225 (diagonal), A189390 (row sums).
Sequence in context: A368260 A368262 A181695 * A184830 A025499 A022474
KEYWORD
nonn,tabl
AUTHOR
Stefano Spezia, Aug 28 2019
STATUS
approved