Exercise 12.

Starting with the generating function for \(1,2,3,4, \ldots\text{,}\) find a generating function for each of the following sequences.

1. \(1, 0, 2, 0, 3, 0, 4,\ldots\text{.}\)

2. \(1, -2, 3, -4, 5, -6, \ldots\text{.}\)

3. \(0, 3, 6, 9, 12, 15, 18, \ldots\text{.}\)

4. \(0, 3, 9, 18, 30, 45, 63,\ldots\text{.}\) (Hint: relate this sequence to the previous one.)