Fix a positive integer $n$. Define sequences $a, b, c \in \mathbb{Q}^{n+1}$ by $(a_0, b_0, c_0) = (0, 0, 1)$ and \[ a_k = (n-k+1) \cdot c_{k-1}, \quad b_k = \binom nk - c_k - a_k, \quad \text{and} \quad c_k = \frac{b_{k-1}}{k} \]for each integer $1 \leq k \leq n$. $ $ $ $ $ $ $ $ $ $ Determine for which $n$ it happens that $a, b, c \in \mathbb{Z}^{n+1}$. Proposed by Jonathan Du