1. In a folk dance, the dancers form two lines facing each other. One line is formed by n women, the other line by n men. Each person joins his or her left hand to either the person opposite him or her, or to the left neighbor, or to the person standing opposite the left neighbor. A similar rule holds for the right hand (change "left" to "right" in the previous rule). No person is allowed to join both hands to the same person. How many ways are there to meet these rules?
Source: Austrian-Polish Mathematical Competition
2. What if the restriction that no person is allowed to join both hands to the same person is dropped?