A394643 - OEIS

%I #33 Apr 01 2026 14:50:49

%S 0,1,7,23,77,198,550,1299,3309,7490,18146,40046,93890,203756,466796,

%T 1001027,2254205,4791306,10646986,22475034,49416630,103744916,

%U 226145492,472651438,1022923602,2130020308,4581964180,9511110524,20353746404,42136635224,89767861592

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

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

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

%H Alois P. Heinz, <a href="/A394643/b394643.txt">Table of n, a(n) for n = 0..3305</a>

%F a(n) = (A057571(n) + A001405(n)*n^2)/2. - _Alois P. Heinz_, Mar 27 2026

%e For n = 3 there are three symmetric Dyck paths of semilength 3 in the first quadrant of the square grid as shown below:

%e     _          _ _        _ _ _

%e      |_           |_           |

%e        |_           |          |

%e          |          |          |

%e .

%e The total number of cells under the symmetric Dyck paths is 6 + 8 + 9 = 23 as shown below:

%e     _          _ _        _ _ _

%e    |_|_       |_|_|_     |_|_|_|

%e    |_|_|_     |_|_|_|    |_|_|_|

%e    |_|_|_|    |_|_|_|    |_|_|_|

%e       6          8          9

%e .

%e So a(3) = 23.

%p b:= proc(x, y) option remember; `if`(y<0 or y>x, 0, `if`(x=0, [1, 0],

%p       add((p-> p+[0, p[1]*(2*y+i)])(b(x-1, y+i)), i=[-1, 1])))

%p     end:

%p a:= n-> (add(b(n, n-2*i), i=0..n/2)[2]+binomial(n, floor(n/2))*n^2)/2:

%p seq(a(n), n=0..30);  # _Alois P. Heinz_, Apr 01 2026

%Y Row sums of A393383.

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

%K nonn,easy

%O 0,3

%A _Omar E. Pol_, Mar 27 2026

%E Terms a(5) and beyond from _Alois P. Heinz_, Mar 27 2026