A098157 Triangle T(n,k) with diagonals T(n,n-k) = binomial(n+1,2k).

1, 1, 1, 0, 3, 1, 0, 1, 6, 1, 0, 0, 5, 10, 1, 0, 0, 1, 15, 15, 1, 0, 0, 0, 7, 35, 21, 1, 0, 0, 0, 1, 28, 70, 28, 1, 0, 0, 0, 0, 9, 84, 126, 36, 1, 0, 0, 0, 0, 1, 45, 210, 210, 45, 1, 0, 0, 0, 0, 0, 11, 165, 462, 330, 55, 1, 0, 0, 0, 0, 0, 1, 66, 495, 924, 495, 66, 1, 0, 0, 0, 0, 0, 0, 13, 286, 1287, 1716, 715, 78, 1

Offset: 0

Author

Paul Barry, Aug 29 2004

Keywords

• easy
• nonn
• tabl

Comments

Row sums are A000079. Diagonal sums are A062200. Inverse is A065547, less the first column.

Number of permutations of length n avoiding simultaneously the patterns 123 and 132 with k descents. A descent in a permutation a(1)a(2)...a(n) is position i such that a(i)>a(i+1). - Tian Han, Nov 16 2023

Examples

			Rows begin:
 {1},
 {1,1},
 {0,3,1},
 {0,1,6,1},
 {0,0,5,10,1},
 {0,0,1,15,15,1},
  ...
		

Links

• T. Han and S. Kitaev, Joint distributions of statistics over permutations avoiding two patterns of length 3, arXiv:2311.02974 [math.CO], 2023.

Crossrefs

Cf. A000079, A062200, A065547.

Programs

Mathematica

Table[Binomial[n+1, 2(n-k)],{n,0,11},{k,0,n}]//Flatten (* Stefano Spezia, Nov 16 2023 *)

Formula

T(n, k) = binomial(n+1, 2(n-k)) with 0 <= k <= n.

G.f.: (1 + x - q*x)/(1 - 2*q*x - q*x^2 + q^2*x^2). - Tian Han, Nov 16 2023