A124428 Triangle, read by rows: T(n,k) = binomial(floor(n/2),k)*binomial(floor((n+1)/2),k).

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, 1, 42, 315, 700, 525, 126, 7, 1, 49, 441, 1225, 1225, 441, 49, 1, 1, 56, 588, 1960, 2450, 1176, 196, 8

Offset: 0

Author

Paul D. Hanna, Oct 31 2006

Keywords

• nonn
• tabf

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

Examples

			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; ...
		

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.

Crossrefs

Cf. A001405 (row sums), A056953, A026003, A124429 (antidiagonal sums), A124430 (eigenvector), A191521.

Columns = A002378, A006011, A006542, etc.

Programs

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

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 *)

PARI

T(n,k)=binomial(n\2,k)*binomial((n+1)\2,k)

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

Formula

A056953(n) = Sum_{k=0..floor(n/2)} k!*T(n,k).

A026003(n) = Sum_{k=0..floor(n/2)} 2^k*T(n,k).