For a reciprocal of a polynomial, $f = \frac{1}{p}$, we (presumably) may construct a sequence $(c_n)_{n=0}^\infty$ such that for all $N\ge 0$
$$f(k)k! = \sum_{n=0}^{N-1} c_n(k-n)! + O((k-N)!). $$
I ask about constructing such sequences in https://mathoverflow.net/questions/378103/calculating-factorial-sequence-of-a-rational-function.

Through some messy calculations, I get $$c_0,c_1,\dotsc = 0,0,0,0,1,-2,7,-32,179,-1182,8993,-77440,744425,-7901410,\dotsc$$ when $f = \frac{1}{(k-1)(k(k-1)(k-2)-k)}$. It just so happens that $\lvert c_{i+3}\rvert = a(i)$ for the OEIS [A265165](http://oeis.org/A265165). I have tried looking at [Banderier, Baril, and Dos Santos - Right-jumps & pattern avoiding permutations][2] 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 the 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?


  [1]: https://mathoverflow.net/questions/378103/calculating-factorial-sequence-of-a-rational-function
  [2]: https://lipn.univ-paris13.fr/~banderier/Papers/rightjump.pdf