A263316 - OEIS

%I #12 Oct 25 2023 09:28:54

%S 1,0,0,1,1,3,7,16,40,98,246,624,1596,4120,10708,28009,73673,194743,

%T 517067,1378365,3687665,9898417,26649117,71943947,194717215,528236599,

%U 1436122339,3912244667,10677558423,29192753795,79944089343,219261036592,602226736360

%N 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 Alois P. Heinz, <a href="/A263316/b263316.txt">Table of n, a(n) for n = 0..1000</a>

%e a(0) = 1: [(0,0)].

%e a(3) = 1: [(0,0),(2,1),(3,3)].

%e a(4) = 1: [(0,0),(3,1),(4,4)].

%e 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 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 a:= proc(n) option remember; `if`(n<5, [1, 0$2, 1$2][n+1],

%p       ((n-3)*a(n-1) +(5*n-5)*a(n-2) +(3*n-3)*a(n-3)

%p        -(4*n-20)*a(n-4) -(4*n-16)*a(n-5))/(n+1))

%p     end:

%p seq(a(n), n=0..40);

%t 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 Table[a[n], {n, 0, 40}] (* _Jean-François Alcover_, Oct 25 2023, after _Alois P. Heinz_ *)

%Y Cf. A014137, A082582, A168592, A218321.

%K nonn

%O 0,6

%A _Alois P. Heinz_, Oct 14 2015