This is a noncommutative version of these three previous questions:

https://mathoverflow.net/questions/80828/differential-operator-power-coefficients

https://mathoverflow.net/questions/337330/%D0%A1losed-formula-for-g-partialn

https://mathoverflow.net/questions/415617/a-leibniz-like-formula-for-fx-fracddxn-fx

Let $(R,\partial)$ be a noncommutative differential ring (unitality does not play a role), i.e.
we have $\forall a,b \in R$:
$$
\partial (ab) = \partial(a) b + a \partial (b).
$$
Fix a generic element ("regular function") $f \in R$. I am interested in an explicit description of the iterations
$$
(R_f \partial)^n f,\ n \in \mathbb{N},
$$
where $R_f$ denotes the multiplication operator by $f$ *from the right*. For example:
$$
(R_f \partial) f = \partial(f) f
$$
and
$$
(R_f \partial)^2 f = (R_f \partial) (\partial(f) f) = \partial^2(f) f^2 + \partial(f)^2 f
$$
My question is thus:
> Is there a known explicit description for the words made out of the letters $\partial^n(f),\dotsc,\partial(f),f$ involved in the expansion of $(R_f \partial)^n f$ and the coefficients in front of these words? Has this been investigated anywhere?

Ideally, I am looking for a description similar to Comtet's theorem in the commutative case cited by [Gjergji Zaimi](https://mathoverflow.net/a/80873) in [the first link mentioned above](https://mathoverflow.net/questions/80828/differential-operator-power-coefficients).

**Aside:** My setting is actually slightly more complicated than this. I only have a derivation "with a twist":
$$
\partial (ab) = \partial(a) \varphi(b) + a \partial(b),
$$
where $\varphi$ is an (injective, non-unital) ring endomorphism with $[\varphi,\partial]$ not being very illuminating, which only seems to complicate the combinatorics even further.