Let $T$ be a function of two complex variables, analytic at the origin. Express $T$ as a power series:
$$T(x,y)=\sum_{i,j}t_{ij}x^iy^j$$
I would like to find a generating function for the sequence $(t_{3m,2m})$.
There are standard tricks for this. One is to write
$$U(x,y)=T(x^2,y^{1/2}/x^3)=\sum_{i,j}t_{ij} x^{2i-3j}y^{j/2}$$ and view this as a power series in $x$. Then the constant term in this power series is $$U_0(y)=\sum_{2i-3j=0}t_{ij}y^{j/2}=\sum_m t_{3m,2m}y^m$$ which is exactly the sought-after generating function.
There are then standard tricks, using residues, to get an explicit form for the function $U_0$.
But even without the residue calculus, one might hope to recover the function $U_0$ by explicitly writing down the function $U$ and then setting $x=0$. For example, take $$T(x,y)={(1+x)(1+y)\over1-xy(2+x+y+xy)}$$ Then $$U(x,y)={x(1+x^2)(x^3+\sqrt{y})\over x^4-2x^3\sqrt{y}-x^5\sqrt{y}-y-x^2y}$$ Setting $x=0$ yields $U_0(x,y)=0$, which is certainly not the right generating function because there exist non-zero values of $t_{3m,2m}$. For example, $t_{3,2}=10$ and $t_{6,4}=81$.
Question: What's wrong here?
I have a suspicion that the problem arises from insufficient care about exactly what rings these functions and power series live in, but when I've tried to take such care, I've still been left baffled.
Since you are taking a square root, the function $U(x,y)$ is not analytic at the origin. And even if it were, you are not expanding in a typical power series, but rather in a kind of Puiseux series (i.e., a power series with fractional exponents).