OFFSET
0,6
COMMENTS
Row sums form A001405, the central binomial coefficients: C(n,floor(n/2)). The eigenvector of this triangle is A124430.
T(n,k) is the number of dispersed Dyck paths of length n (i.e., Motzkin paths of length n with no (1,0) steps at positive heights) having k peaks. Example: T(5,2)=3 because, denoting U=(1,1), D=(1,-1), H=(1,0), we have HUDUD, UDHUD, and UDUDH. - Emeric Deutsch, Jun 01 2011
From Emeric Deutsch, Jan 18 2013: (Start)
T(n,k) is the number of Dyck prefixes of length n having k peaks. Example: T(5,2)=3 because we have (UD)(UD)U, (UD)U(UD), and U(UD)(UD); the peaks are shown between parentheses.
T(n,k) is the number of Dyck prefixes of length n having k ascents and descents of length >= 2. Example: T(5,2)=3 because we have (UU)(DD)U, (UU)D(UU), and (UUU)(DD); the ascents and descents of length >= 2 are shown between parentheses. (End)
T(n,k) is the number of noncrossing partitions of [n] having n-k blocks, such that the nontrivial blocks are of type {a,b}, with a < = n/2 and b > n/2. Such partitions have k nontrivial blocks, uniquely determined by the choice of k first elements among floor(n/2) elements, and the choice of k second elements among floor((n+1)/2) elements. Indeed, by planarity, any two blocs {a,b} and {c,d} satisfy a < c iff b > d. - Francesca Aicardi Nov 03 2022
LINKS
G. C. Greubel, Rows n = 0..100 of triangle, flattened
Jean-Luc Baril, Alexander Burstein, and Sergey Kirgizov, Pattern statistics in faro words and permutations, arXiv:2010.06270 [math.CO], 2020. See paragraph 2.1.
FORMULA
EXAMPLE
Triangle begins:
1;
1;
1, 1;
1, 2;
1, 4, 1;
1, 6, 3;
1, 9, 9, 1;
1, 12, 18, 4;
1, 16, 36, 16, 1;
1, 20, 60, 40, 5;
1, 25, 100, 100, 25, 1;
1, 30, 150, 200, 75, 6;
1, 36, 225, 400, 225, 36, 1; ...
MATHEMATICA
Table[Binomial[Floor[n/2], k]*Binomial[Floor[(n+1)/2], k], {n, 0, 15}, {k, 0, Floor[n/2]}]//Flatten (* G. C. Greubel, Feb 24 2019 *)
PROG
(PARI) T(n, k)=binomial(n\2, k)*binomial((n+1)\2, k)
(Magma) [[Binomial(Floor(n/2), k)*Binomial(Floor((n+1)/2), k): k in [0..Floor(n/2)]]: n in [0..15]]; // G. C. Greubel, Feb 24 2019
(Sage) [[binomial(floor(n/2), k)*binomial(floor((n+1)/2), k) for k in (0..floor(n/2))] for n in (0..15)] # G. C. Greubel, Feb 24 2019
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Paul D. Hanna, Oct 31 2006
STATUS
approved