This question concerns Kummers confluent Hypergeometric function $_1F_1$, sometimes denoted $M$. Recall that $$ M(a;b;z) = {_1F_1 (a;b;z)} = \sum_{n=0}^{\infty} \frac{(-1)^n}{n!}\frac{(a)_n}{(b)_n} $$
I am seeking to approximate $M(-\lfloor \alpha X\rfloor;-X;-1)$ when $X$ is a large positive real and $\alpha \in [0,1]$, perhaps by way of a series in $\alpha$.
Numerically, I find that:
$$ M(-\lfloor \alpha X\rfloor;-X;-1) \approx 1 - \left( \frac{e-1}{e}+\frac{\pi}{10}\right)\alpha + \left( \frac{\pi}{10}\right) \alpha^2 $$ is within 0.4% of the true value when $X\gg100$ and $\alpha \in [0,1]$. However, I see no combinatorial interpretation, or way to derive this from the hypergeometric series, or general approximation to higher orders.