A375763 Irregular triangle read by rows, T(n,k) is the number of North-East lattice paths from (0,0) to (n,n+2) that stay weakly above y = x, with weight = k + A000217(n).

1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 3, 4, 5, 4, 4, 3, 2, 1, 1, 1, 4, 7, 10, 11, 11, 11, 9, 8, 6, 5, 3, 2, 1, 1, 1, 5, 11, 18, 24, 27, 30, 29, 28, 25, 23, 19, 16, 12, 10, 7, 5, 3, 2, 1, 1, 1, 6, 16, 30, 46, 59, 71, 78, 81, 81, 78, 74, 67, 60, 52, 46, 37, 31, 24

Offset: 0

Author

John Tyler Rascoe, Aug 26 2024

Keywords

• nonn
• easy
• tabf

Comments

Here the weight of a lattice path is the area under the path and above the x-axis. T(n,k) also counts the number of integer compositions of (3*n) + (2*k) + 6 with adjacent differences in {-1,1}, first part 1, and last part 3.

Examples

			Triangle begins:
    k=0  1  2   3   4   5   6   7   8   9  10  11  12  13  14
 n=0: 1;
 n=1: 1, 1, 1;
 n=2: 1, 2, 2,  2,  1,  1;
 n=3: 1, 3, 4,  5,  4,  4,  3,  2,  1,  1;
 n=4: 1, 4, 7, 10, 11, 11, 11,  9,  8,  6,  5,  3,  2,  1,  1;
 ...
T(1,0) = 1: (NENN).
T(2,1) = 2: (NNEENN) and (NENNEN).
T(3,2) = 4: (NENENNNE), (NENNENEN), (NNEENNEN), and (NNENEENN).
		

Links

• John Tyler Rascoe, Python program.

Crossrefs

Cf. A000245 (empirical row sums), A000217 (row lengths).

Cf. A227543 (paths of this kind from (0,0) to (n,n), offset 1 for (0,0) to (n,n+1)).

Cf. A000108, A152659, A173258, A227543, A268429.

Programs

Python

# see linked program