2020 Putnam B4

Let $n$ be a positive integer, and let $V_n$ be the set of integer $(2n+1)$-tuples $\mathbf{v} = (s_0, s_1, \cdots, s_{2n-1}, s_{2n})$ for which $s_0 = s_{2n} = 0$ and $|s_j - s_{j-1}| = 1$ for $j=1,2,\cdots,2n$. Define

$$

q(\mathbf{v}) = 1 + \sum_{j=1}^{2n-1} 3^{s_j},

$$

and let $M(n)$ be the average of $\frac{1}{q(\mathbf{v})}$ over all $\mathbf{v} \in V_n$. Evaluate $M(2020)$.

令$n$为正整数，并令$V_n$为整数$(2n+1)$元组的集合$\mathbf{v} = (s_0, s_1, \cdots, s_{2n-1}, s_{2n})$，其中$s_0 = s_{2n} = 0$且$|s_j - s_{j-1}| = 1$ 对于 $j=1,2,\cdots,2n$。定义

$$

q(\mathbf{v}) = 1 + \sum_{j=1}^{2n-1} 3^{s_j},

$$

令 $M(n)$ 为 $\frac{1}{q(\mathbf{v})}$ 对所有 $\mathbf{v} \in V_n$ 的平均值。评估 $M(2020)$。