A333114 Sum over all closed Deutsch paths of length n of products over all peaks p of x_p/y_p, where x_p and y_p are the coordinates of peak p.

1, 0, 1, 1, 5, 11, 44, 134, 529, 1902, 7793, 31068, 133641, 574259, 2594969, 11842726, 56083004, 269450143, 1333170844, 6703500545, 34548749471, 181026885253, 969167994094, 5273977173249, 29257773480987, 164894374634333, 945779302210358, 5507572390808676

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.

Examples

			a(4) = (1/1)*(3/1) + 2/2 + 3/3 = 5.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..800
• Helmut Prodinger, Deutsch paths and their enumeration, arXiv:2003.01918 [math.CO], 2020
• Wikipedia, Counting lattice paths

Crossrefs

Cf. A005043, A005411, A330169, A333098.

Programs

Maple

b:= proc(x, y, t) option remember; `if`(x=0, 1, add(
      `if`(t and j<0, x/y, 1)*b(x-1, y+j, is(j>0)), j=[
      `if`(y=0, [][], -1), $1..x-1-y]))
    end:
a:= n-> b(n, 0, false):
seq(a(n), n=0..30);

Mathematica

b[x_, y_, t_] := b[x, y, t] = If[x == 0, 1, Sum[If[t && j < 0, x/y, 1]* b[x-1, y+j, j > 0], {j, Join[If[y == 0, {}, {-1}], Range[x-1-y]]}]];
a[n_] := b[n, 0, False];
a /@ Range[0, 30] (* Jean-François Alcover, Mar 19 2020, after Alois P. Heinz *)