Putnam 1975

Let $n = 2m$, where $m$ is an odd integer greater than 1. Let $\theta = e^{2\pi i / n}$. Express $(1-\theta)^{-1}$ explicitly as a polynomial in $\theta$,

$a_k \theta^k + a_{k-1} \theta^{k-1} + \cdots + a_1 \theta + a_0$,

with integer coefficients $a_i$.

[Note that $\theta$ is a primitive $n$-th root of unity, and thus it satisfies all of the identities which hold for such roots.]

Let $f_0(x) = e^x$ and $f_n(x) = xf'_n(x)$ for $n = 0, 1, 2, \dots$ Find \sum_{n=0}^\infty \frac{f_n(1)}{n!} in terms of e.