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Mathematica can actually solve the recursion relation in closed form, $$F_n(n)=-\tfrac{1}{2}(n^2-1)^{-1}\bigl[\left(\frac{n-1}{n}\right)^n \left(n \left(\frac{n}{n-1}\right)^n \Phi \left(\frac{n}{n-1},1,n+1\right)+n (n+2) \left(\frac{n}{n-1}\right)^n \Phi \left(\frac{n}{n-1},2,n+1\right)-(n-1) \left((n+2) \text{Li}_2\left(\frac{n}{n-1}\right)+3 n-\log \left(\frac{1}{1-n}\right)+6\right)\right)\bigr].$$ The large-$n$ limit then evaluates to $$\lim_{n\rightarrow\infty} F_n(n)=\frac{18+\pi^2}{12 e}.$$