expect doing 还是expect to do

都可，但意思上有差别：

表示时间持续的，可以用doing，例：
I expect your coming back.
我（一直）期待着你归来，强调时间上的持续。

而to do，例：
I expect to win the game.
我期望在比赛中获胜，强调对结果的期望。

The officer expected his men to do their duty.
The chairman of the committee expect to be back next Thursday.

expect + to do.
look forward to +doing

expect to do...
expect that...