Closed formula for the coefficients of a series obtained from an expansion.

Question

The Heisenberg algebra is generated by $h_i, i\in \mathbb{Z}\backslash\{0\}$ and the central element $c$. We expand the function $$\exp (\sum_{n=1}^{\infty}h_{-n}\frac{z^n}{n}) = 1 + \sum_{n=1}^{\infty}h_{-n}\frac{z^n}{n} + (\sum_{n=1}^{\infty}h_{-n}\frac{z^n}{n})^2+\cdots \\ = 1 + h_{-1} z + (\frac{h_{-2} + \frac{h_{-1}^2}{2}}{2}) z^2 +(\frac{h_{-3}}{3}+\frac{h_{-1}h_{-2}}{4} + \frac{h_{-2}h_{-1}}{4} + \frac{h_{-1}^3}{6})z^3 + \\ + (\frac{h_{-4}}{4}+\frac{h_{-1}h_{-3}}{6}+\frac{h_{-2}h_{-2}}{8}+\frac{h_{-3}h_{-1}}{6}+ \\ + \frac{h_{-1}^2h_{-2}}{6}+\frac{h_{-2}h_{-1}^2}{6}+\frac{h_{-1}h_{-2}h_{-1}}{6}+\frac{h_{-1}^4}{24})z^4 + \cdots .$$ Denote $$ \exp (\sum_{n=1}^{\infty}h_{-n}\frac{z^n}{n}) = \sum_{n=0}^{\infty} P_nz^n. $$ Then $$P_0=1, P_1=h_{-1}, P_2 = \frac{h_{-2} + \frac{h_{-1}^2}{2}}{2}, \ldots.$$ Is there a close formula for $P_i, i \in \mathbb{Z}_{\geq 0}$? Or is there a recurrence formula which can compute $P_i$? Are there some references about this problem? Thank you very much.