This is a noncommutative version of these three previous questions:

differential operator power coefficients

Сlosed formula for $(g\partial)^n$

A Leibniz-like formula for $(f(x) \frac{d}{dx})^n f(x)$?

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 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 $\partial^n(f),\dots,\partial(f),f$ 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 in the first link mentioned above.

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.

This is a noncommutative version of these three previous questions:

differential operator power coefficients

Сlosed formula for $(g\partial)^n$

A Leibniz-like formula for $(f(x) \frac{d}{dx})^n f(x)$?

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 in the first link mentioned above.

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.