Finding generating function

Question

Find generating function for

$1$, $\frac{5k(k+1)}{2}$, $\frac{25k(k+1)(k+2)}{6}$, $\frac{125k(k+1)(k+2)(k+3)}{24}$, $\dots$

I can't even start. I don't know where to start. Please someone explain.

Answer 1

It seems that $$x_n = \frac{5^n}{n!}\prod_{i=0}^n(k+i)$$ for $n$ from 0.