Difficult# Pascal's Triangle Row Products

COMBIN-EY9GSN

Let:

$$P_n=\prod_{k=0}^n {n\choose k}$$

...denote the product of all elements in row $n$ of Pascal's Triangle. What is the value of the limit as $n$ goes to $\infty$ of the fraction:

$$\frac{P_{n+1}P_{n-1}}{\left(P_n\right)^2}?$$