Let $(F(x)\frac{d}{dx})^n=\sum\_{i=1}^n H\_{n,i}(F, F', F^{(2)}, \ldots , F^{(n)})\frac{d^i}{dx^i}$. I'm curious about the exact formula for $H\_{n,i}(y\_0, y\_1, \ldots , y\_n)$. What is known about it?



differential operator power coefficients
Let me start by giving the formula


$$H\_{n,l}(y\_0,y\_1,\dots,y\_n)=\sum\_{(k\_1,k\_2,\dots,k\_{n-1})\in P\_{n,l}}\frac{y\_{0}}{l!}\prod\_{j=1}^n (j+1-k\_1-\cdots-k\_j)\frac{y\_{k\_j}}{k\_j!}$$


where, $P\_{n.l}$ is the set of $(k\_1,k\_2,\dots,k\_{n-1})\in \mathbb Z\_{\geq 0}^{n-1}$ which satisfy $k\_1+\cdots+k\_{n-1}=n-l$ and $k\_1+\cdots+k\_i\le i$ for all $1\le i \le n$.


In this form this is due to L. Comtet:



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> L. Comtet, [Une formule explicite pour les puissances successives de l'opérateur de dérivation de Lie](https://gallica.bnf.fr/ark:/12148/bpt6k6217213f/f179.item), C. R. Acad. Sci. Paris Sér. A-B 276 (1973) A165–A168. [Zbl 0252.05002](https://zbmath.org/0252.05002)
> 
> 
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This has an OEIS entry, [A139605](https://oeis.org/A139605), where you can find more references. In particular see Bergeron, F. and Reutenauer, C., [Une interprétation combinatoire des puissances d'un opérateur différentiel linéaire](http://www.labmath.uqam.ca/%7Eannales/volumes/11-2/PDF/269-278.pdf "zbMATH review at https://zbmath.org/0635.05019"), Ann. Sci. Math. Quebec. 11, 269–278 (1987) for a combinatorial interpretation in terms of forests of rooted trees.


The analogous expansion for the multivariable case is treated in ["Universal expansion of the powers of a derivation"](https://doi.org/10.1007/BF00750066 "Lett Math Phys 34, 343–364 (1995). zbMATH review at https://zbmath.org/0837.05010") by M. Ginocchio. There are also generalizations to the non-commutative case that are described in his other article ["On the Hopf algebra of functional graphs and differential algebras"](https://doi.org/10.1016/S0012-365X(97)00081-2 "Discrete Math. 183, No. 1–3, 119–140 (1998). zbMATH review at https://zbmath.org/0928.05060").


