For a reciprocal of a polynomial, $f = \frac{1}{p}$, we (presumably) may construct a sequence $(c_n)_{n=0}^\infty$ such that $$f(k)k! = \sum_{n=0}^{N-1} c_n(k-n)! + O((k-N)!). $$I ask about constructing such sequences in this question.

Through some messy calculations, I get $$c_0,c_1,\dots = 0,0,0,0,1,-2,7,-32,179,-1182,8993,-77440,744425,-7901410,\dots$$ when $f = \frac{1}{(k-1)(k(k-1)(k-2)-k)}$. It just so happens that $|c_{i+3}| = a(i)$ for the OEIS A265165. I have tried looking at this paper which the OEIS page links, but I couldn't see a reason why these should yield the same sequence. Can you find one? (funnily enough, I came across this sequence $c_n$ by considering permutations, but in a totally different setting than this paper)

Also, are "factorial sequences" $(c_n)$ a concept which has been studied before? Are there any other interesting OEIS sequences that happen to have a factorial sequence?