A358492 Irregular triangle read by rows: T(n,k) is one half of the number of line segments of length 1 in the k-th antidiagonal of the Dyck path described in the n-th row of A237593.

1, 1, 1, 1, 2, 1, 1, 2, 1, 2, 2, 1, 1, 1, 3, 1, 1, 3, 2, 1, 1, 1, 3, 2, 1, 1, 1, 3, 3, 1, 1, 1, 1, 4, 2, 1, 1, 1, 4, 2, 2, 1, 1, 1, 1, 1, 3, 4, 1, 1, 1, 1, 3, 4, 2, 1, 1, 1, 1, 2, 4, 2, 2, 1, 1, 1, 1, 1, 3, 5, 2, 1, 1, 1, 1, 1, 1, 3, 5, 2, 1, 1, 1, 1, 1, 3, 5, 2, 2, 1, 1, 1, 1, 1, 1, 1, 5, 4, 2, 1, 1, 1, 1, 1, 1, 5, 4, 2, 2

Offset: 1

Author

Omar E. Pol, Nov 19 2022

Keywords

• nonn
• tabf

Examples

			Triangle begins (first 19 rows):
  1;
  1, 1;
  1, 2;
  1, 1, 2;
  1, 2, 2;
  1, 1, 1, 3;
  1, 1, 3, 2;
  1, 1, 1, 3, 2;
  1, 1, 1, 3, 3;
  1, 1, 1, 1, 4, 2;
  1, 1, 1, 4, 2, 2;
  1, 1, 1, 1, 1, 3, 4;
  1, 1, 1, 1, 3, 4, 2;
  1, 1, 1, 1, 2, 4, 2, 2;
  1, 1, 1, 1, 1, 3, 5, 2;
  1, 1, 1, 1, 1, 1, 3, 5, 2;
  1, 1, 1, 1, 1, 3, 5, 2, 2;
  1, 1, 1, 1, 1, 1, 1, 5, 4, 2;
  1, 1, 1, 1, 1, 1, 5, 4, 2, 2;
...
For n = 10 the 10th row of A237593 is [6, 2, 1, 1, 1, 1, 2, 6]. When that row is interpreted as a symmetric Dyck path in the fourth quadrant using 20 line segments of length 1 the Dyck path looks like this:
.
                         |
                         |
                         |
                         |
                         |
                      _ _|
                    _|
                  _|
                 |
      _ _ _ _ _ _|
.
The numbers of line segments of length 1 in the successive antidiagonals are respectively [2, 2, 2, 2, 8, 4] so the 10th row of triangle is [1, 1, 1, 1, 4, 2].
		

Crossrefs

Row sums give A000027.

Row n has length A008619(n).

Column 1 gives A000012.

Cf. A196020, A235791, A236104, A237270, A237271, A237591, A237593, A245092, A262626, A339575.