Who's up to the challenge? 74

$\large \lim _{ n\to \infty }{ (f(n))^{ 1/2n^{ 2 } } }$

Let $f(n)$ denote the number of perfect matchings which cover the $2n\times 2n$ square planar lattice. If the limit above can be expressed in the form $A{ e }^{ BG/\pi }$ for positive integers $A$ and $B$ . Find the product $AB$ .

Notation : $\displaystyle G = \sum_{n=0}^\infty \dfrac{ (-1)^n}{(2n+1)^2} \approx 0.916$ is the Catalan's constant .