All these problems require some facility working with infinite series, or in some cases just finite series. For example, consider the first problem. The ordinary power series generating function is, by definition,
It looks like it would pay to factor out x^3:
I don't remember a closed form for
but it reminds me of
Other series likely to prove useful (so you should know them) are the Binomial Theorem and the infinite series for e^x.
If you have a textbook, it's likely to list a some useful infinite series. If not, you can probably find a math handbook with some series that might come in handy.
Hope this helps--