A394643 Total number of cells under all symmetric Dyck paths of semilength n in the first quadrant of the square grid.

0, 1, 7, 23, 77, 198, 550, 1299, 3309, 7490, 18146, 40046, 93890, 203756, 466796, 1001027, 2254205, 4791306, 10646986, 22475034, 49416630, 103744916, 226145492, 472651438, 1022923602, 2130020308, 4581964180, 9511110524, 20353746404, 42136635224, 89767861592

Offset

0,3

Comments

The total number of cells equals the total area under all symmetric Dyck paths.

Equivalently, total number of cells above all symmetric Dyck paths of semilength n in the fourth quadrant of the square grid.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..3305

Formula

a(n) = (A057571(n) + A001405(n)*n^2)/2. - Alois P. Heinz, Mar 27 2026

Example

For n = 3 there are three symmetric Dyck paths of semilength 3 in the first quadrant of the square grid as shown below:
    _          _ _        _ _ _
     |_           |_           |
       |_           |          |
         |          |          |
.
The total number of cells under the symmetric Dyck paths is 6 + 8 + 9 = 23 as shown below:
    _          _ _        _ _ _
   |_|_       |_|_|_     |_|_|_|
   |_|_|_     |_|_|_|    |_|_|_|
   |_|_|_|    |_|_|_|    |_|_|_|
      6          8          9
.
So a(3) = 23.

Maple

b:= proc(x, y) option remember; `if`(y<0 or y>x, 0, `if`(x=0, [1, 0],
      add((p-> p+[0, p[1]*(2*y+i)])(b(x-1, y+i)), i=[-1, 1])))
    end:
a:= n-> (add(b(n, n-2*i), i=0..n/2)[2]+binomial(n, floor(n/2))*n^2)/2:
seq(a(n), n=0..30);  # Alois P. Heinz, Apr 01 2026

Crossrefs

Row sums of A393383.

Cf. A000290, A001405, A057571, A390502, A394529.

Sequence in context: A356684 A048539 A240526 * A018886 A145842 A086908

Adjacent sequences: A394639 A394640 A394642 * A394644 A394645 A394646

Keyword

nonn,easy

Author

Omar E. Pol, Mar 27 2026

Extensions

Terms a(5) and beyond from Alois P. Heinz, Mar 27 2026

Status

approved