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

0,5

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..800

Helmut Prodinger, Deutsch paths and their enumeration, arXiv:2003.01918 [math.CO], 2020

Wikipedia, Counting lattice paths

Example

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

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 *)

Crossrefs

Cf. A005043, A005411, A330169, A333098.

Sequence in context: A141355 A262020 A222368 * A276300 A222476 A097056

Adjacent sequences: A333111 A333112 A333113 * A333115 A333116 A333117

Keyword

nonn

Author

Alois P. Heinz, Mar 07 2020

Status

approved