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