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Calculus Level 5

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

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.

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