A300323 Number of Dyck paths of semilength n such that the area under the right half of the path equals the area under the left half of the path.

1, 1, 2, 3, 6, 12, 28, 69, 186, 522, 1536, 4638, 14408, 45568, 146884, 479871, 1589516, 5320854, 18000198, 61412376, 211282386, 731973720, 2553168136, 8957554412, 31604599044, 112060048354, 399227283950, 1428315878002, 5130964125124, 18499652813682

Offset: 0

Author

Alois P. Heinz, Mar 02 2018

Keywords

• nonn

Examples

			              /\
             /  \      /\/\
a(3) = 3:   /    \    /    \    /\/\/\ .
.
a(5) = 12 counts A001405(5) = 10 symmetric plus 2 non-symmetric Dyck paths:
             /\  /\
          /\/  \/  \  and its reversal.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..300
• Wikipedia, Counting lattice paths

Crossrefs

Column k=0 of A300322.

Cf. A000108 (all Dyck paths), A000225, A001405 (symmetric Dyck paths), A129182, A239927, A298645.

Programs

Maple

b:= proc(x, y) option remember; expand(`if`(x=0, 1,
      `if`(y<1,   0, b(x-1, y-1)*z^(2*y-1))+
      `if`(x add(coeff(p, z, i)^2
      , i=0..degree(p)))(b(n, n-2*j)), j=0..n/2)
    end:
seq(a(n), n=0..32);

Mathematica

b[x_, y_] := b[x, y] = Expand[If[x == 0, 1, If[y < 1, 0, b[x - 1, y - 1] z^(2y - 1)] + If[x < y + 2, 0, b[x - 1, y + 1] z^(2y + 1)]]];
a[n_] := a[n] = Sum[Function[p, Sum[Coefficient[p, z, i]^2, {i, 0, Exponent[p, z]}]][b[n, n - 2j]], {j, 0, n/2}];
Table[a[n], {n, 0, 32}] (* Jean-François Alcover, May 31 2018, from Maple *)

Formula

a(n) >= A001405(n) with equality only for n <= 4.

a(n) is odd <=> n in { A000225 }.