Let $\langle a_{k,n} : k,n \in \mathbb{N} \rangle$ be a bivariate number sequence, and let $f(x,y) = \sum_k \sum_n a_{k,n} x^k y^n$ be its corresponding bivariate generating function. Is there a method for obtaining the generating function $g(x)$ of the central coefficients $\langle a_{k,k} \rangle$ directly from $f(x,y)$?
By "direct method" I mean something that does not involve expanding $f$ and taking the sub-sequence $\langle a_{k,k} \rangle$. My first idea was to make $y=x$ in $f(x,y)$, but that doesn't work.
