A308113 - OEIS

%I #18 Jan 02 2021 15:01:53

%S 1,1,1,2,7,22,68,205,634,2011,6490,21178,69785,231940,776794,2618951,

%T 8881373,30274185,103673227,356500914,1230497234,4261633997,

%U 14805279769,51580807121,180173390369,630864082719,2213834486422,7784823272163,27427361186479

%N 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.

%H Alois P. Heinz, <a href="/A308113/b308113.txt">Table of n, a(n) for n = 0..550</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Lattice_path#Counting_lattice_paths">Counting lattice paths</a>

%F a(n) ~ c * d^n / n^(3/2), where d = 3.7137893481485186502229788321701955452444... and c = 0.47404607017890475336081188752626598456... - _Vaclav Kotesovec_, May 24 2019

%p b:= proc(x, y) option remember; `if`(y=0, 1, add(add(`if`(x+v

%p       >y+h or igcd(h, v)>1, 0, b(x-h, y-v)), v=1..y), h=1..x))

%p     end:

%p a:= n-> b(n$2):

%p seq(a(n), n=0..30);

%t 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}]];

%t a[n_] := b[n, n];

%t a /@ Range[0, 30] (* _Jean-François Alcover_, Jan 02 2021, after _Alois P. Heinz_ *)

%Y Cf. A308087, A308114.

%K nonn

%O 0,4

%A _Alois P. Heinz_, May 13 2019