Mathematics Question from JEE Main 2023 · 30 January, Shift 1

Let $\sum_{n=0}^{\infty} \frac{n^3 \big( (2n)! \big) + (2n-1)(n!)}{(n!)(2n)!} = a e + \frac{b}{e} + c,$ where $a, b, c \in \mathbb{Z}$ and $e = \sum_{n=0}^{\infty} \frac{1}{n!}$. Then $a^2 - b + c$ is equal to _____.