The Lucas property of a number array Original Research Article

For all nonnegative integers i,j let w(i,j | a,b,c) denote the number of all paths in the plane from (0,0) to (i,j) with steps (1,0), (0,1), (1,1), and with positive integer weights a, b, c, respectively. The divisibility property of the array w(i,j | a,b,c) is studied. The notation of the Lucas property is introduced. Let p be a prime and let w̄(i,j | a,b,c) denote the remainders of dividing w(i,j | a,b,c) by p where 0⩽w̄(i,j | a,b,c)