A393383 Irregular triangle read by rows in which row n lists the number of cells under the symmetric Dyck paths of semilength n in the first quadrant of the square grid in nondecreasing order, n >= 0.

0, 1, 3, 4, 6, 8, 9, 10, 11, 12, 13, 15, 16, 15, 17, 17, 18, 19, 20, 21, 22, 24, 25, 21, 22, 23, 23, 24, 24, 25, 26, 26, 27, 27, 28, 28, 29, 30, 31, 32, 33, 35, 36, 28, 30, 30, 30, 31, 32, 32, 32, 33, 33, 34, 34, 34, 35, 35, 36, 36, 37, 37, 37, 38, 38, 39, 39, 40, 40, 41, 41, 42, 43, 44, 45, 46, 48, 49

Offset

0,3

Comments

Both the column 1 and the right border are the same as in its supersequence A390502.

The number of cells equals the area under the symmetric Dyck path.

Equivalently row n lists the number of cells above the symmetric Dyck paths in fourth quadrant.

The diagram in the fourth quadrant of the square grid is the same as the Young diagram corresponding to a partition of T(n,k).

Note that at least one of the terms in the row n equals A024916(n), the sum of all divisors of all positive integers <= n, because the associated symmetric Dyck path is described in the n-th row of A237593.

Links

Alois P. Heinz, Rows n = 0..17, flattened

Example

Triangle begins:
   0;
   1;
   3,  4;
   6,  8,  9;
  10, 11, 12, 13, 15, 16;
  ...
For n = 3 there are three symmetric Dyck paths of semilength 3 in the first quadrant of the square grid as shown below:
    _         _ _       _ _ _
     |_          |_          |
       |_          |         |
         |         |         |
.
The illustration of the third row of the triangle is as shown below:
    _         _ _       _ _ _
   |_|_      |_|_|_    |_|_|_|
   |_|_|_    |_|_|_|   |_|_|_|
   |_|_|_|   |_|_|_|   |_|_|_|
      6         8         9
.

Maple

b:= proc(x, y) option remember; `if`(y<0 or y>x, 0, `if`(x=0, 1,
       expand(b(x-1, y-1)*z^(2*y-1)+ b(x-1, y+1)*z^(2*y+1))))
    end:
T:= n-> (p-> seq((i+n^2)/2$coeff(p, z, i), i=0..degree(p)))(
             add(b(n, n-2*i), i=0..n/2)):
seq(T(n), n=0..7);  # Alois P. Heinz, Apr 01 2026

Crossrefs

Subsequence of A390502.

Column 1 gives A000217.

Row lengths give A001405.

Right border gives A000290.

Row sums give A394643.

Cf. A024916, A088855, A109077, A109078, A237593, A339575.

Sequence in context: A099356 A377431 A325209 * A121543 A080702 A156167

Adjacent sequences: A393380 A393381 A393382 * A393384 A393385 A393386

Keyword

nonn,tabf,look

Author

Omar E. Pol, Mar 26 2026

Extensions

More terms from Alois P. Heinz, Mar 27 2026

Status

approved