Prove that  

$$\lim\_{n\to\infty}\Bigg[\sum\_{i,j}{2i\choose i}{2j\choose j}\Bigg]^{1/n}=4$$ 
such that, $i+j=n$
and $0\le i,j \le n$.

My idea was to first sum the series when $i<j$ and when $j<i$. But I am unable to proceed. Please enlighten me.