If you are talking about grammar in bigger sentences... you can say stuff like "I expect no pity from you." or "I expect no thanks from you." If you say I expect not it is either wrong or really old fashioned sounding. The most common way to negate expect is "I don't expect" or "I do not expect" or "I didn't expect" etc...

"I expect not" is fine. "I expect no" is not quite correct, instead say "I expect that the answer will be no."