A366920 a(n) is the number times a Dyck path in an m X m box of any size has area n, counted to the lower right.

1, 1, 0, 1, 1, 0, 1, 2, 1, 1, 1, 3, 3, 3, 2, 2, 5, 6, 7, 7, 5, 6, 8, 12, 15, 18, 16, 16, 15, 17, 24, 32, 40, 43, 45, 45, 42, 44, 53, 69, 87, 104, 115, 126, 125, 124, 124, 136, 160, 198, 240, 282, 321, 345, 360, 365, 367, 382, 417, 482, 574, 682, 791, 895, 976

Offset: 0

Author

William J. Keith, Oct 28 2023

Keywords

• nonn

Comments

A Dyck path in an m X m grid is a set of up steps U and right steps R from the lower left corner to the upper right corner, staying weakly above the diagonal.

For this statistic, count the boxes below and to the right of the path.

The first time an area appears in two different squares is at size 15, which appears in the 4 X 4 box below UUURURRR and in the 5 X 5 box below URURURURUR.

Examples

			The 0 X 0 box yields the trivial (empty) path of area 0.
The 1 X 1 box yields one Dyck path of area 1 (UR).
The 2 X 2 box yields one Dyck path each of area 3 (URUR) and 4 (UURR).
The 3 X 3 box yields one Dyck path of area 6 (URURUR), two of area 7 (UURRUR and URUURR), and one each of area 8 (UURURR) and 9 (UUURRR).
		

Crossrefs

Cf. A143951, A227543.

Formula

G.f.: 1 + q + q^3 + q^4 + q^6 + 2q^7 + ...

To construct the g.f., take A(x,q) as defined in A227543, and replace each instance of x^k with q^(k*(k+1)/2).

Extensions

a(45)-a(65) from Alois P. Heinz, Oct 29 2023