A339575 Triangle read by rows: T(n,k) is the number of cells in the k-th row of the diagram constructed in the fourth quadrant with the Dyck path described in the n-th row of A237593, 1 <= k <= n.

1, 2, 2, 3, 3, 2, 4, 4, 4, 3, 5, 5, 5, 3, 3, 6, 6, 6, 6, 5, 4, 7, 7, 7, 7, 5, 4, 4, 8, 8, 8, 8, 8, 6, 5, 5, 9, 9, 9, 9, 9, 7, 7, 5, 5, 10, 10, 10, 10, 10, 10, 8, 7, 6, 6, 11, 11, 11, 11, 11, 11, 8, 7, 6, 6, 6, 12, 12, 12, 12, 12, 12, 12, 10, 10, 9, 7, 7

Offset: 1

Author

N. J. A. Sloane, Dec 11 2020

Keywords

• nonn
• tabl
• look

Comments

These are the lengths of the rows if one regards the n-th region in the diagram as the Young diagram corresponding to a partition of A024916(n).

Column k gives the partial sums of the k-th column of triangle A240061. - Omar E. Pol, Dec 11 2020

Examples

			Triangle begins:
   1;
   2,  2;
   3,  3,  2;
   4,  4,  4,  3;
   5,  5,  5,  3,  3;
   6,  6,  6,  6,  5,  4;
   7,  7,  7,  7,  5,  4,  4;
   8,  8,  8,  8,  8,  6,  5,  5;
   9,  9,  9,  9,  9,  7,  7,  5,  5;
  10, 10, 10, 10, 10, 10,  8,  7,  6,  6;
  11, 11, 11, 11, 11, 11,  8,  7,  6,  6,  6;
  12, 12, 12, 12, 12, 12, 12, 10, 10,  9,  7,  7;
...
From _Omar E. Pol_, Jan 19 2022: (Start)
For n = 10 the Dyck path described in the 10th row of A237593 is as shown below in the fourth quadrant:
.
   k                                     10th row
      . . . . . . . . . . . . . X       of triangle
   1  .                    |                10
   2  .                    |                10
   3  .                    |                10
   4  .                    |                10
   5  .                    |                10
   6  .                 _ _|                10
   7  .               _|                     8
   8  .             _|                       7
   9  .            |                         6
  10  . _ _ _ _ _ _|                         6
      .
      .
     -y
.
T(10,k) is the number of cells in the k-th row of the diagram.
The total number of cells in all rows of the diagram is equal to A024916(10) = 87, the same as the sum of the 10th row of triangle. (End)
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 1..10011 (rows for n = 1..141, flattened)
• Rémy Sigrist, PARI program for A339575

Crossrefs

Row sums are A024916.

Cf. A000203, A236104, A237270, A237591, A237593, A240061.

Programs

PARI

See Links section.

Extensions

Name edited by Omar E. Pol, Jan 22 2022