Let me start by giving first the formula
$$H_{n,l}(y_0,y_1,\dots,y_n)=\sum_{(k_1,k_2,\dots,k_{n-1})\in P_{n,l}}\frac{y_{0}}{l!}\prod_{j=1}^n (j+1-k_1-\cdots-k_j)\frac{y_{k_j}}{k_j!}$$
which in this form 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, where you can find more references. In particular I find these notes of T. Copeland have a nice graphic description in terms of rooted trees (though some people may prefer to think of this in terms of a Bratteli diagram of partitions with appropriate weights, but it amounts to the same thing).