A258181 Sum over all Dyck paths of semilength n of products over all peaks p of 2^(x_p-y_p), where x_p and y_p are the coordinates of peak p.

1, 1, 5, 89, 5933, 1540161, 1584150165, 6497470064169, 106497075348688637, 6980195689972655145233, 1829876050804408046228327525, 1918781572083632396857805205324025, 8047973452254281276702044410544321359565, 135022681866797995009325363468217320506328688097

Offset

0,3

Comments

A Dyck path of semilength n is a (x,y)-lattice path from (0,0) to (2n,0) that does not go below the x-axis and consists of steps U=(1,1) and D=(1,-1). A peak of a Dyck path is any lattice point visited between two consecutive steps UD.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..55

Wikipedia, Lattice path

Formula

a(n) ~ c * 2^(n*(n-1)), where c = 1.47818066525747143617276638534... . - Vaclav Kotesovec, Jun 01 2015

Maple

b:= proc(x, y, t) option remember; `if`(y>x or y<0, 0,
      `if`(x=0, 1, b(x-1, y-1, false)*`if`(t, 2^(x-y), 1) +
                   b(x-1, y+1, true)  ))
    end:
a:= n-> b(2*n, 0, false):
seq(a(n), n=0..15);

Mathematica

b[x_, y_, t_] := b[x, y, t] = If[y > x || y < 0, 0, If[x == 0, 1, b[x - 1, y - 1, False]*If[t, 2^(x - y), 1] + b[x - 1, y + 1, True]]];
a[n_] := b[2*n, 0, False];
Table[a[n], {n, 0, 15}] (* Jean-François Alcover, Apr 23 2016, translated from Maple *)

Crossrefs

Cf. A000108, A000698, A005411, A005412, A258172, A258173, A258174, A258175, A258176, A258177, A258178, A258179, A258180.

Sequence in context: A028353 A191512 A015085 * A355082 A067258 A167424

Adjacent sequences: A258178 A258179 A258180 * A258182 A258183 A258184

Keyword

nonn

Author

Alois P. Heinz, May 22 2015

Status

approved