Find the inverse of the following lower triangular Toeplitz matrix
$$\mathbf{A}_{M\times M}=\left[\begin{array}{ccccc} 1\\ -a_{1} & 1\\ -a_{2} & -a_{1} & 1\\ \vdots & & & \ddots\\ -a_{M-1} & \cdots & -a_{2} & -a_{1} & 1 \end{array}\right], $$
where $$a_i=\frac{c\Gamma(i-c)}{\Gamma(i+1)\Gamma(1-c)}$$ and $c\in(0,1)$ is a real constant.
I hope to find the closed-form expression of $\Vert \mathbf{A} ^{-1} \Vert_1$, where $\Vert\cdot\Vert_1$ represents the $L_1$ induced norm. (Actually, in this case, it is the summation of the first column of $\mathbf{A}^{-1}$.) And if possible, I hope to find how the matrix size $M$ affects the value of $\Vert \mathbf{A} ^{-1} \Vert_1$, i.e., $$ f(M)=\Vert \mathbf{A}_{M\times M} ^{-1} \Vert_1. $$ Even a tight approximation or bounds are acceptable.
There are some properties of $\{a_i\}$, for example, $\sum_{i=1}^{\infty}a_i=1$. But by now I haven't had ideas how to solve it. I appreciate if anyone could help me.