Delannoy Numbers

Number of paths from (0,0) to (n,n) in an n X n grid using only right,top or top-right. 


When you will try to find no of paths from (0,0) to (p,q) in a grid where you can move only right,top or top-right.

Some Results are




K(p,1)=2p+1 (for p>0)(3,5,7,9,....)K(p,2)=p^2+(p-1)^2 (for p>0) (5,13,25,41....)


Now when you try to find the K(p,q) when p=q you will get a sequence like this 1, 3, 13, 63, 321, 1683, 8989.... Which is known as  Delannoy Numbers.

 It has been Proved that nTh Secquence of Delannoy Number is equal to Sum_{k=0..n} C(n,k)*C(n+k,k)