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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.

%I A308113 #18 Jan 02 2021 15:01:53
%S A308113 1,1,1,2,7,22,68,205,634,2011,6490,21178,69785,231940,776794,2618951,
%T A308113 8881373,30274185,103673227,356500914,1230497234,4261633997,
%U A308113 14805279769,51580807121,180173390369,630864082719,2213834486422,7784823272163,27427361186479
%N 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.
%H A308113 Alois P. Heinz, <a href="/A308113/b308113.txt">Table of n, a(n) for n = 0..550</a>
%H A308113 Wikipedia, <a href="https://en.wikipedia.org/wiki/Lattice_path#Counting_lattice_paths">Counting lattice paths</a>
%F A308113 a(n) ~ c * d^n / n^(3/2), where d = 3.7137893481485186502229788321701955452444... and c = 0.47404607017890475336081188752626598456... - _Vaclav Kotesovec_, May 24 2019
%p A308113 b:= proc(x, y) option remember; `if`(y=0, 1, add(add(`if`(x+v
%p A308113       >y+h or igcd(h, v)>1, 0, b(x-h, y-v)), v=1..y), h=1..x))
%p A308113     end:
%p A308113 a:= n-> b(n$2):
%p A308113 seq(a(n), n=0..30);
%t A308113 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 A308113 a[n_] := b[n, n];
%t A308113 a /@ Range[0, 30] (* _Jean-François Alcover_, Jan 02 2021, after _Alois P. Heinz_ *)
%Y A308113 Cf. A308087, A308114.
%K A308113 nonn
%O A308113 0,4
%A A308113 _Alois P. Heinz_, May 13 2019