Is Barack Obama a natural born citizen? His father was not a US citizen.
Barack Obama (assuming that he was born in Hawaii, something that I do not accept as fact for many reasons that go far beyond published information) then he is a citizen by birth as the citizenship bequeathed to him his mother would be recognized under those specific circumstances. If he were born in Kenya then he would not have been recognized as a US Citizen by birth.
I do not accept that there is a class of super-citizen by birth called natural born citizen. I believe that the terms are effectively synonymous. I base this view on a close reading of all the contemporaneous writings on the subject that were available to the Framers.