= Solution
The only <pole> is the simple one at $z=1$, with \b[$\boxed{\operatorname{Res}_{1}=e}$]. The point $z=0$ is an <essential singularity>, not another <pole>: multiplying $e^{1/z^2}$ by a nonvanishing analytic factor cannot remove its infinite principal part. For completeness its <residue> can also be read from
$$
-\left(\sum_{j\geq0}z^j\right)\left(\sum_{k\geq0}\frac{z^{-2k}}{k!}\right),\qquad0<|z|<1.
$$
The coefficient of $z^{-1}$ is $-\sum_{k\geq1}1/k!=1-e$. This extra <residue> belongs to the <essential singularity>; it does not change the list of <poles>.
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