Constructing PDA for $a^{2n} b^{3n}$
The following pushdown automaton should do the trick. I publish this only because the existing answer can be improved upon. (Note, I am using $e$ to denote $\epsilon$- (or $\lambda$-) transitions.



The idea is that the left-hand part counts the number of $a$'s (modulo 2).
Each time it has seen two $a$'s, it pushes $3$ $b$'s onto the stack. Nondeterministically, the machine can change to the right-hand state. It then matches a $b$ from the string fro each $b$ on the stack.