  The reciprocal of $e^{-x}$ has a power series about $0$ in which all
coefficients are non-negative. Gessel [Reciprocals of exponential polynomials
and permutation enumeration, Australas. J. Combin., 74, 2019] considered
truncates of the power series of $e^{-x}$, i.e. polynomials of the form
$\sum_{n=0}^r (-1)^n\frac{x^n}{n!}$, and established combinatorially that the
reciprocal of the truncate has a power series with all coefficients
non-negative precisely when $r$ is odd.
  Here we extend Gessel's observations to arbitrary ``thinned exponential
series''.
  To be precise, let $A \subseteq \{1,3,5,\ldots\}$ and $B \subseteq
\{2,4,6,\ldots\}$, and consider the series
  \[
  1-\sum_{a \in A} \frac{x^a}{a!} + \sum_{b \in B} \frac{x^b}{b!}.
  \]
  We consider conditions on $A$ and $B$ that ensure that the reciprocal series
has all coefficients non-negative. We give combinatorial proofs for a large set
of conditions, including whenever $1 \in A$ and the endpoints of the maximal
consecutive intervals in $A \cup B$ are odd integers.
  In particular, the coefficients in the reciprocal series can be interpreted
as ordered set partitions of $[n]$ with block size restrictions, or in terms of
permutations with restricted lengths of maximally increasing runs, suitably
weighted.
