I want to evaluate at $t=1$ the inverse Laplace Transform $\mathcal{L}^{-1}\{F(s)\}\vert_{t=1}$ of
$$ F(s) = \frac{1}{s^2-\sum\limits_{n=1}^\infty{n!s^{3-n}x^n}} $$
and find out the $x^n$ coefficient $[x^n]\mathcal{L}^{-1}\{F(s)\}\vert_{t=1}$ of this evaluation.
If I approximate the sum to $m$ terms I can use Mathematica to calculate the coefficients of $x^n$ to be $1, 1, 2, 4, 23/3, 14,\ldots$.
However, I am after an analytic expression for $\mathcal{L}^{-1}\{F(s)\}\vert_{t=1}$ or $[x^n]\mathcal{L}^{-1}\{F(s)\}\vert_{t=1}$.
Is this feasible?