Find the generating function for given series

Question

I have this given assignment, and I need a couple of hints to get me started since I don't know how to do this. what is the generating function for the given series: $\{ 1,3,5,7,9 \}$?

Answer 1

There are many sequences whose first five terms are 1, 3, 5, 7 and 9. As an example, consider $$a(n) = 5 - \frac{n}{20} (n - 3) (n - 6) ((n - 3)^2 + 4) \text{ for } n=1, \ldots, 5.$$ For other possibilities, you can check The On-Line Encyclopedia of Integer Sequences, which has 400+ results for this, including the popular a(n) = 2n+1 similar to the one mentioned in the comments.