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 [Calculating "factorial sequence" of a rational function](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](https://lipn.univ-paris13.fr/%7Ebanderier/Papers/rightjump.pdf) 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?



Why does this "factorial sequence" appear in the OEIS?
We have
$$\frac{1}{x}f(\frac1x) = \sum\_{n\geq 1} (F\_{2n-3}-1)x^n,$$
where $F\_{2n+1}$ are Fibonacci numbers.
Per [answers](https://mathoverflow.net/a/378165) to [your previous question](https://mathoverflow.net/questions/378103/calculating-factorial-sequence-of-a-rational-function), it follows that
$$c\_{n+1} = \sum\_{i=0}^n s(n,i) (F\_{2i-3}-1).$$


For $n\geq 2$, "$-1$" can be dropped, reducing the formula to
$$c\_{n+1} = \sum\_{i=0}^n s(n,i) F\_{2i-3}.$$


Using the generating function for Stirling numbers and Binet's formula for Fibonacci numbers, we obtain the following exponential generating function for $c\_{n+1}$:
$$\sum\_{n\geq 0} c\_{n+1} \frac{z^n}{n!} = \frac{(1+z)^{1+\phi}(2\phi-3)+(1+z)^{2-\phi}(2\phi+1)}{\sqrt{5}}-1,$$
where $\phi\mathrel{:=}\frac{1+\sqrt{5}}2$. Under substitution of $z=-x$, this matches the e.g.f. given in the sequence [OEIS A265165](https://oeis.org/A265165), and so they do represent the same sequence (up to signs and shift of indices).


