Delannoy numbers D(m,n): paths (0,0)→(m,n) using R=(1,0), U=(0,1), D=(1,1). More paths than just R,U because diagonals are allowed. D(m,n) = Σ C(m,k)C(n,k)2^k. Central Delannoy D(n,n): 1,3,13,63,321,1683...
How to Use
Enter m and n:
- D(m,n): Path count
- Triangle: Full grid
- Central: D(n,n) sequence
Closed Form
D(m,n) = Σ_{k=0}^{min(m,n)} C(m,k)·C(n,k)·2^k. The 2^k factor accounts for diagonal steps being 'worth' two moves. Also: D(m,n) = Σ C(m+n-k, m)·C(m,k) using Vandermonde-like identities.
Connections
Central Delannoy D(n,n) = Σ C(n,k)². Schröder numbers are related: S(n) = D(n,n) − D(n-1,n). Delannoy numbers also appear in queen moves on a chessboard and mathematical physics.
Step-by-Step Instructions
- 1Enter m and n.
- 2Compute D(m,n).
- 3View grid.
- 4Central sequence.
- 5Compare.