Let \( a = 1 + \frac{{}^{2}C_{2}}{3!} + \frac{{}^{3}C_{2}}{4!} - \frac{{}^{4}C_{2}}{5!} \ldots \), \( b = 1 + \frac{{}^{1}C_{0} + {}^{1}C_{1}}{1!} + \frac{{}^{2}C_{0} + {}^{2}C_{1} + {}^{2}C_{2}}{2!} + \frac{{}^{3}C_{0} + {}^{3}C_{1} + {}^{3}C_{2} + {}^{3}C_{3}}{3!} + \ldots \). Then \( \dfrac{2b}{a^{2}} \) is equal to ________.
