We have a generating function enumerating some objects indexed by two variables of the form $$ F(x, y) = \sum_{i \geq0,\ j \geq 0} f(i, j)\ x^iy^j $$ which is symmetric, i.e $f(i, j) = f(j, i)$.
Let us then consider a generating function whose coefficients correspond to fixing one of the parameters and letting the other vary. $$ G_j(x) = \sum_{i \geq 0} f(i, j)\ x^i $$
If we know that $G_j(x)$ is a rational function for every fixed $j$, can we say anything about $F(x, y)$? For example, can we say if it is algebraic? D-finite?
For example, consider the case where $f(i,j) = 0$ for $i \ne j$. Then $G_j(x) = f(j,j) x^j$ is a monomial for each $j$, but $F(x,y) = \sum_i f(i,i) (x y)^i$ is an arbitrary series in powers of $xy$.