Limit of a Sum Involving Binomial Coefficients I would like to prove that $$\dfrac{\sqrt{n}}{{{2n \choose n}^2}} \cdot \sum_{j=0}^n {n \choose j}^4$$ converges to $\sqrt{\dfrac{2}{\pi}}$ as $n \to \infty$.
Evaluating the sum in Matematica for large values of $n$ suggests that $\sqrt{\dfrac{2}{\pi}}$ is indeed the correct limit.

It is enough to use the Central Limit Theorem. For large values of $n$, the binomial distribution converges to a normal distribution, and by Stirling's approximation
$$ \binom{2n}{n}\sim\frac{4^n}{\pi n}$$
holds, hence your limit is just the reciprocal of:
$$ \int_{-\infty}^{+\infty}e^{-2x^2}\,dx = \sqrt{\frac{\pi}{2}}.$$
