Factorial Series I

Question

Given the factorial series \begin{align} \beta_{1}(x) = \sum_{s=1}^{\infty} \frac{(-1)^{s} \, s!}{x(x+1)(x+2)\cdots(x+s)} \end{align} then what are the coefficients, $a_{s}$, of the square of $\beta_{1}(x)$, \begin{align} [ \beta_{1}(x)]^{2} = \sum_{s=1}^{\infty} \frac{s! \, a_{s}}{x(x+1)(x+2) \cdots (x+s)}. \end{align} I speculate that the coefficients are a finite series of the form \begin{align} a_{m} = \sum_{r=1}^{m} b_{r}. \end{align}