Mathematica can actually solve the recursion relation in closed form,
$$F_n(n)=-\tfrac{1}{2}(n^2-1)^{-1}\left(\frac{n-1}{n}\right)^n\left[n \left(\frac{n}{n-1}\right)^n \Phi \left(\frac{n}{n-1},1,n+1\right)+n (n+2) \left(\frac{n}{n-1}\right)^n \Phi \left(\frac{n}{n-1},2,n+1\right)-(n-1) \left((n+2) \text{Li}_2\left(\frac{n}{n-1}\right)+3 n+\ln(1-n) +6\right)\right],$$
with $\Phi$ the <A HREF="https://dlmf.nist.gov/25.14">Lerch transcendent</A> and $\text{Li}_2$ the polylog.
The large-$n$ limit then evaluates to
$$\lim_{n\rightarrow\infty} F_n(n)=\frac{18+\pi^2}{12 e}.$$