Enumeration of Random Walk Positions in \(L_1\)-norm ball in \(\mathbb{Z}^d\)

In this paper, we mainly concerned about deriving the general formula to count the possible positions of \(n\) step random walk in \(\mathbb{Z}^d\) with unit length in each step, which we denoted as \(|P_n^{d}|\). For our results, we firstly propose a recurrence relation of the counting formula: \(|P_n^{d+1}| = |P_n^d| + 2\sum_{k=0}^{n-1} |P_k^d|\). Next, we propose two methods in deriving the explicit formula of \(|P_n^{d}|\) using generating functions and Faulhaber's formula. Finally, we reached our main theorem in the matrix representation of our formula.