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.