How to prove:
$$\Gamma(-n+x)= \frac{(-1)^n}{n!}\left( \frac{1}{x}+\left(-\gamma + \sum_{k=1}^{n}\frac{1}{k} \right) +\frac{1}{6} \left(\pi^2+3 \left(-\gamma + \sum_{k=1}^{n}\frac{1}{k} \right)^2 +3 \left(-\frac{\pi^2}{6}+ \sum_{k=1}^n\frac{1}{k^2} \right) \right)x + \mathcal{O}(x^2) \right)$$
with integer $n > 0$.