Divisibility Properties of Some Number Arrays

For all nonnegative integers \(i,j\), let \(q(i, j)\) denote the number of all lattice paths in the plane from \((0,0)\) to \((i, j)\) with steps \((1,0)\), \((0,1)\), and \((1,1)\). In this paper, it is proved that

\[q(i_{n}p^{n}+…+i_0,j_np^n+…+j_0)\equiv(i_n,j_n)…q(i_0,j_0) \pmod{p}\]

where \(p\) is an odd prime and \(0 \leq i_k < p\), \(0 \leq j_k < p\). This relation implies a remarkable pattern to the divisibility of the array of numbers \(q(i, j)\).