Let $B,C\,\colon\mathbb N^2\to\mathbb R$ be functions satisfying $B(n,k)=C(n,k)=0$ whenever $k>n$. Suppose we know the generating functions $$F_B(x)=\sum_{n\geq 0}B(n,k)x^n$$ and $$F_C(x)=\sum_{n\geq 0}C(n,k)x^n$$ for any fixed $k$. Is there a general method one might use to determine information (specifically, asymptotic estimates) about the numbers $$A(n):=\sum_{k=0}^n B(n,k)C(n,k)?$$
My initial thought was to define a bivariate generating function $$G(x,y):=\sum_{n\geq 0}\sum_{k\geq 0}B(n,k)C(n,k)x^ny^k$$ and note that $G(x,1)$ is the ordinary generating function for $A(n)$. However, I'm not quite sure how to proceed from there.
In particular, I'm interested in the case $$B(n,k)={n+k-1\choose k\,,\,2k-1\,,\,n-2k}$$ and $$C(n,k)={2n-k\choose n}.$$ Any asymptotic estimates for the numbers $A(n)$ in this specific case would be greatly appreciated.