The condition to be checked here is no. of b's following a's should be more than the no. of a's but less than twice the no. of b's. This cannot be done using a DPDA. But for each a we can non-deterministically guess that it can generate either one b or two bs and this way we can make a PDA. The context-free grammar would be
$$S \to aSb \mid aSbb \mid \epsilon$$