Let the sequence $(a_n)_{n \geq 1}$ be defined by $a_1 >0 $, and the recurrent relation $$a_n = 2a_{n-1}*\frac{(n-1)^{(n-1)}}{(n-1)!}$$
Then, what is $\lim_{n\to\infty} \frac {{a_{n+1}}^{1/{n+1}}}{{a_n}^{1/n}} $?
So far, I've managed to prove that $\lim_{n\to\infty}{(\frac{a_{n+1}}{a_n})}^{1/n}=e,$ by using the Stirling limit. Therefore, it should suffice to calculate $\lim_{n\to\infty} {{a_{n+1}}^{\frac{-1}{n(n+1)}}}. $
A linear recurrence of the first order, homogeneous. Easy to see that the solution is:
$$ a_n = a_1 2^ n \prod_{1 \le k \le n} \frac{k^k}{k!} $$
Compute the limit from here. Much of the mess should simplify away.