Is Yehuda Hanasi descended from the royal family?

The rabbis tell us that Rabbi Yehuda Hanasi and Hillel were descended from King David. Their royal lineage earned them respect with the Roman government and the office of Nasi remained in the family of Hillel for over 400 years. But there are contradictions between the Babylonian and Jerusalem Talmuds and Midrashim about their lineage. In this class (given at Five Blocks) I try to resolve and explain these contradictions.