A224776 Number of lattice paths from (0,0) to (n,n) that do not go below the x-axis or above the diagonal x=y and consist of steps D=(1,-1), H=(1,0) and S=(0,1).
1, 1, 3, 14, 83, 568, 4271, 34296, 288946, 2524676, 22695611, 208713400, 1955285936, 18601484936, 179267898087, 1746795785272, 17183086302528, 170427862676296, 1702621483524154, 17118538010217472, 173092651634957516, 1759113081143064184, 17959329720442879275
Offset: 0
Keywords
Examples
a(0) = 1: the empty path. a(1) = 1: HS. a(2) = 3: HSHS, HHSS, HSDSS. a(3) = 14: HSHSHS, HHSSHS, HSDSSHS, HSHHSS, HHSHSS, HSDSHSS, HHHSSS, HSDHSSS, HSHDSSS, HHSDSSS, HSDSDSSS, HSHSDSS, HHSSDSS, HSDSSDSS.
Links
- Alois P. Heinz, Table of n, a(n) for n = 0..960
Crossrefs
Programs
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Maple
b:= proc(x, y) option remember; `if`(y>x, 0, `if`(x=0, 1, b(x-1, y)+`if`(y>0, b(x, y-1), 0)+b(x-1, y+1))) end: a:= n-> b(n, n): seq(a(n), n=0..25); -
Mathematica
b[x_, y_] := b[x, y] = If[y > x, 0, If[x == 0, 1, b[x - 1, y] + If[y > 0, b[x, y - 1], 0] + b[x - 1, y + 1]]]; a[n_] := b[n, n]; a /@ Range[0, 25] (* Jean-François Alcover, Dec 18 2020, after Alois P. Heinz *)
Formula
a(n) ~ c * ((11+5*sqrt(5))/2)^n / n^(3/2), where c = 0.01403940208697420741365874329992235342402687... . - Vaclav Kotesovec, Sep 07 2014