For $n \in \mathbb{N}$ a given dimension and $m \geq 1$, consider the lower triangular matrix $A(m)\in \mathbb{R}_+^{n\times n}$ defined by
$$A_{i,j}(m) := \dfrac{1}{m+i}\dfrac{1}{m-1+i} \quad \text{ if } i \leq j $$
$$A_{i,j}(m) := 0 \quad \text{ if } i \geq j+1 $$
I am interested in understanding the behavior of the coefficients $\mathbf{x}^{(M)} \in \mathbb{R}^n$ given by
$$ \mathbf{x}^{(M)} := \left(\prod\limits_{m=1}^M A(m)\right) \mathbf{1} $$
where $\mathbf{1} \in \mathbb{R}^n$ is a column vector of 1's.
I seek sharp bounds on the values of $\mathbf{x}^{(M)}$, I have the explicit relations that $\mathbf{x}^{(1)}_j = \dfrac{j}{j+1}$ and $\mathbf{x}^{(M)}_1 = 1/(M+1)!M!$
However is there an explicit generalized relation? If not, then what bounds, if any, can be constructed?