SCHWEITZER 2021 Problem 9

For a given natural number $n$ , two players randomly (uniformly distributed) select a common number $0 \le j \le n$ , and then each of them independently randomly selects a subset of $\{1,2, \cdots, n \}$ with $j$ elements. Let $p_n$ be the probability that the same set was chosen. Prove that $$\sum_{k=1}^{n} p_k = 2 \log{n} + 2 \gamma - 1 + o(1), (n \to \infty),$$ where $\gamma$ is the Euler constant.