I want to construct an entire function with one zero at $0$, and zeros at negative natural numbers with degree $n$ (multiplicity $n$ at $z = -n$ for $n \geq 1$).

Does $f(z) = z\displaystyle\prod_{n=1}^{\infty}(z+n)^{-n}$ work?

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Probably one should look at the shape of Weierstrass products, and the discussion of their convergence, to get oriented here. –  paul garrett Sep 24 '12 at 13:40