Is the sequence $a_n = \sum_{k=0}^n \frac{1}{k!} \int_0^{n-k+1} x^k e^{-x} dx$ bounded?

Let$$a_n = \sum_{k=0}^n \frac{1}{k!} \int_0^{n-k+1} x^k e^{-x} dx$$for all $n \ge 0$.

Is the sequence $(a_n)_{n \ge 0}$ bounded?

The integral $\int_0^{n-k+1} x^k e^{-x} dx$ is the lower incomplete gamma function, which converges to $k!$ as $n \to \infty$. Hence, we trivially have$$a_n \le \sum_{k=0}^n \frac{1}{k!} \cdot k! = n,$$but the right-hand side is unbounded. Is it possible to obtain a better estimate?