A333098 Number of closed Deutsch paths whose area is exactly n.

1, 1, 1, 2, 4, 6, 11, 21, 36, 64, 117, 208, 371, 669, 1197, 2141, 3844, 6888, 12336, 22119, 39644, 71034, 127323, 228200, 408955, 732957, 1313647, 2354298, 4219447, 7562249, 13553161, 24290307, 43533784, 78022169, 139833177, 250612596, 449153751, 804984038

Offset: 0

Author

Alois P. Heinz, Mar 07 2020

Keywords

• nonn

Comments

Deutsch paths (named after their inventor Emeric Deutsch by Helmut Prodinger) are like Dyck paths where down steps can get to all lower levels. Open paths can end at any level, whereas closed paths have to return to the lowest level zero at the end.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1250
• Helmut Prodinger, Deutsch paths and their enumeration, arXiv:2003.01918 [math.CO], 2020
• Wikipedia, Counting lattice paths

Crossrefs

Cf. A330169.

Programs

Maple

b:= proc(x, y, k) option remember; `if`(x=0, `if`(y=0
      and k=0, 1, 0), `if`(k2*x*y+x^2-x-y, 0,
      add(b(x-1, y-j, k-(2*y-j)), j=[-1, $1..y])))
    end:
a:= n-> add(b(x, 0, 2*n), x=0..2*n):
seq(a(n), n=0..40);

Mathematica

b[x_, y_, k_] := b[x, y, k] = If[x == 0, If[y == 0 && k == 0, 1, 0], If[k < x || k > 2x y + x^2 - x - y, 0, Sum[b[x - 1, y - j, k - (2y - j)], {j, Join[{-1}, Range[y]]}]]];
a[n_] := Sum[b[x, 0, 2n], {x, 0, 2n}];
a /@ Range[0, 40] (* Jean-François Alcover, Mar 12 2020, after Alois P. Heinz *)