The goal of this problem is to prove the irrationality of a particular real number in a specific way.

Let $P_0(x), \ldots, P_k(x)$ be polynomials with integer coefficients, where $P_k(n) \neq 0$ for $n \geq k$. Let $\alpha = (\alpha_0, \ldots, \alpha_{k-1})$ and $\beta = (\beta_0, \ldots, \beta_{k-1})$ be tuples of integers.

Let $(a_n)$ and $(b_n)$ denote the sequences defined by the recurrence
$$P_k(n)u_n = P_{k-1}(n)u_{n-1} + \cdots + P_0(n)u_{n-k}$$
for $n \geq k$, when initialized with $\alpha$ and $\beta$ respectively.

Let $L_n$ denote $\operatorname{lcm}(1,2,\ldots,n)$ and let $B_n = (L_n)^m b_n$, where $m$ is a fixed positive integer.

Let $X$ be a real number. It is known that if the following hold, then $X$ is irrational.

1. $a_n$ is an integer for all $n \geq 0$.

2. $B_n$ is an integer for all $n \geq 0$.

3. $X \neq b_n / a_n$ for all $n \geq 0$.

4. $\lvert b_n - a_n X\rvert \leq C\rho^n e^{-mn}$ for all $n \geq 0$, where $C$ and $\rho$ are positive real numbers and $\rho < 1$.

Your task is to find $P_i$, $\alpha$, $\beta$, and $m$ that establish the irrationality of ___ in this way.