I obtained this list of series (there are more but listed up to 5th order) and I suspect they are related to the Laguerre polynomials. Strictly speaking they are not $L_n^{\alpha}(x)$ but something close.
Is there a way to figure out the parameterization? It may be the derivatives or integrals of $L_n^{\alpha}(x)$. Any hints are welcome!
As epi163sqrt's link demonstrates, the polynomials are
$$P_n(x) = \sum_{k=0}^n \binom nk \frac{(n - k+1)!}{n!}(-x)^k$$ Which simplifies to $$P_n(x) = \sum_{k=0}^n \frac{n-k+1}{k!}(-x)^k$$
If we define $E_n(x) = \sum_{k=0}^n \frac{x^k}{k!}$ to be the $n$-th order Taylor polynomial for $e^x$, then $$P_n(x) = (n+1)E_n(-x) + xE_{n-1}(-x)$$