A308113 Number of lattice paths from (0,0) to (n,n) that do not go above the diagonal x=y and consist of steps (h,v) with min(h,v) > 0 and gcd(h,v) = 1.

1, 1, 1, 2, 7, 22, 68, 205, 634, 2011, 6490, 21178, 69785, 231940, 776794, 2618951, 8881373, 30274185, 103673227, 356500914, 1230497234, 4261633997, 14805279769, 51580807121, 180173390369, 630864082719, 2213834486422, 7784823272163, 27427361186479

Offset: 0

Author

Alois P. Heinz, May 13 2019

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..550
• Wikipedia, Counting lattice paths

Crossrefs

Cf. A308087, A308114.

Programs

Maple

b:= proc(x, y) option remember; `if`(y=0, 1, add(add(`if`(x+v
      >y+h or igcd(h, v)>1, 0, b(x-h, y-v)), v=1..y), h=1..x))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..30);

Mathematica

b[x_, y_] := b[x, y] = If[y == 0, 1, Sum[Sum[If[x + v > y + h || GCD[h, v] > 1, 0, b[x - h, y - v]], {v, 1, y}], {h, 1, x}]];
a[n_] := b[n, n];
a /@ Range[0, 30] (* Jean-François Alcover, Jan 02 2021, after Alois P. Heinz *)

Formula

a(n) ~ c * d^n / n^(3/2), where d = 3.7137893481485186502229788321701955452444... and c = 0.47404607017890475336081188752626598456... - Vaclav Kotesovec, May 24 2019