cp's OEIS Frontend

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.

%I A366920 #29 Dec 05 2023 20:11:56
%S A366920 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,
%T A366920 24,32,40,43,45,45,42,44,53,69,87,104,115,126,125,124,124,136,160,198,
%U A366920 240,282,321,345,360,365,367,382,417,482,574,682,791,895,976
%N 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.
%C A366920 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.
%C A366920 For this statistic, count the boxes below and to the right of the path.
%C A366920 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.
%F A366920 G.f.: 1 + q + q^3 + q^4 + q^6 + 2q^7 + ...
%F A366920 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).
%e A366920 The 0 X 0 box yields the trivial (empty) path of area 0.
%e A366920 The 1 X 1 box yields one Dyck path of area 1 (UR).
%e A366920 The 2 X 2 box yields one Dyck path each of area 3 (URUR) and 4 (UURR).
%e A366920 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).
%Y A366920 Cf. A143951, A227543.
%K A366920 nonn
%O A366920 0,8
%A A366920 _William J. Keith_, Oct 28 2023
%E A366920 a(45)-a(65) from _Alois P. Heinz_, Oct 29 2023