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: 
>L. Comtet, [Une formule explicite pour les puissances successives de l'opérateur de dérivation de Lie][1], C. R. Acad. Sci. Paris Sér. A-B 276 (1973) A165–A168. [Zbl 0252.05002][2]

This has an OEIS entry, [A139605][3], where you can find more references. In particular see Bergeron, F. and Reutenauer, C., [Une interprétation combinatoire des puissances d'un opérateur différentiel linéaire][4], 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"][5] 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"][6]. 


  [1]: https://gallica.bnf.fr/ark:/12148/bpt6k6217213f/f179.item
  [2]: https://zbmath.org/0252.05002
  [3]: https://oeis.org/A139605
  [4]: http://www.labmath.uqam.ca/~annales/volumes/11-2/PDF/269-278.pdf "zbMATH review at https://zbmath.org/0635.05019"
  [5]: https://doi.org/10.1007/BF00750066 "Lett Math Phys 34, 343–364 (1995). zbMATH review at https://zbmath.org/0837.05010"
  [6]: 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"