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A263316 Number of lattice paths from (0,0) to (n,n) which do not go above the diagonal x=y using steps (1,k), (k,1) with k>=2.

%I A263316 #12 Oct 25 2023 09:28:54
%S A263316 1,0,0,1,1,3,7,16,40,98,246,624,1596,4120,10708,28009,73673,194743,
%T A263316 517067,1378365,3687665,9898417,26649117,71943947,194717215,528236599,
%U A263316 1436122339,3912244667,10677558423,29192753795,79944089343,219261036592,602226736360
%N A263316 Number of lattice paths from (0,0) to (n,n) which do not go above the diagonal x=y using steps (1,k), (k,1) with k>=2.
%H A263316 Alois P. Heinz, <a href="/A263316/b263316.txt">Table of n, a(n) for n = 0..1000</a>
%e A263316 a(0) = 1: [(0,0)].
%e A263316 a(3) = 1: [(0,0),(2,1),(3,3)].
%e A263316 a(4) = 1: [(0,0),(3,1),(4,4)].
%e A263316 a(5) = 3: [(0,0),(3,1),(4,3),(5,5)], [(0,0),(2,1),(4,2),(5,5)], [(0,0),(4,1),(5,5)].
%e A263316 a(6) = 7: [(0,0),(2,1),(3,3),(5,4),(6,6)], [(0,0),(2,1),(4,2),(5,4),(6,6)], [(0,0),(4,1),(5,4),(6,6)], [(0,0),(4,1),(5,3),(6,6)], [(0,0),(3,1),(5,2),(6,6)], [(0,0),(2,1),(5,2),(6,6)], [(0,0),(5,1),(6,6)].
%p A263316 a:= proc(n) option remember; `if`(n<5, [1, 0$2, 1$2][n+1],
%p A263316       ((n-3)*a(n-1) +(5*n-5)*a(n-2) +(3*n-3)*a(n-3)
%p A263316        -(4*n-20)*a(n-4) -(4*n-16)*a(n-5))/(n+1))
%p A263316     end:
%p A263316 seq(a(n), n=0..40);
%t A263316 a[n_] := a[n] = If[n < 5, {1, 0, 0, 1, 1}[[n+1]], ((n-3)a[n-1] + (5n-5)a[n-2] + (3n-3)a[n-3] - (4n-20)a[n-4] - (4n-16)a[n-5])/(n+1)];
%t A263316 Table[a[n], {n, 0, 40}] (* _Jean-François Alcover_, Oct 25 2023, after _Alois P. Heinz_ *)
%Y A263316 Cf. A014137, A082582, A168592, A218321.
%K A263316 nonn
%O A263316 0,6
%A A263316 _Alois P. Heinz_, Oct 14 2015