Asymptotics of a Sequence Defined from Multinomial Coefficients

Question

I am trying to obtain asymptotic estimates for a sequence $A(n)$ defined by $$A(n)=\sum_{k=1}^n\frac{2k}{(n-k+1)(n+k)}{2n-k\choose n}{n+k\choose k\,,\,2k-1\,,\,n-2k+1}.$$ I'm not very familiar with asymptotics in combinatorics, so I don't really know if this is a manageable problem or if there is an easy solution that don't know about. Any help would be appreciated.