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, C. R. Acad. Sci. Paris Sér. A-B 276 (1973) A165–A168

This has an OEIS entry, <a href="http://oeis.org/A139605">A139605</a>, where you can find more references. In particular see Bergeron, F. and Reutenauer, C., Une interpretation combinatoire des puissances d'un operateur differentiel lineaire, 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 <a href="http://www.springerlink.com/content/g58q46214245p42m/">"Universal expansion of the powers of a derivation"</a> by M. Ginocchio. There are also generalizations to the non-commutative case that are described in his other article <a href="http://www.sciencedirect.com/science/article/pii/S0012365X97000812">"On the Hopf algebra of functional graphs and differential algebras"</a>.