Construct a function with zero at $z=0$ and zeros at $z=-n$ with multiplicities $n$.
My answer is $$f(z) = z\prod_{n=1}^{\infty}\left[E_n\left(-\frac zn\right)\right]^n,$$ where $E_n(z)=(1-z)\exp\left(\sum_{k=1}^n\frac{z^k}k\right)$.
Is that right? And does the product converge?