How to find an explicit formula from a generating function

Question

I'm very n00b in maths, and I don't understand what I can do with a generating function.
For example, if I want to compute the first spread polynomial, how can I do that from the generating function:

or the exponential one:

?

Thanks in advance, rubik

Answer 1

If you know what Taylor expansion is, then you should know that $S_n(s)$ in e.g.f. is just its $n$th derivative at zero. For the ordinary g.f., it's almost the same, except that you have to divide the derivative by $n!$.

Anyway, if you want to do more with g.f. than this, I strongly advise you to at least look at this wikipedia page and perhaps even into the generatingfunctionology book, which in my opinion does a pretty good job of introducing g.f. to a beginner.

Answer 2

HINT $\quad\quad $ With $\rm\ \ \alpha\ =\ 1 - 2\ s + 2\ \sqrt{s^2 - s}\ \ $ we have

$$ \rm \frac{s\ x\ (1+x)}{(1-x)^3 + 4\ s\ x\ (1-x)}\ =\ \frac{\alpha}{4\ (x-\alpha)} + \frac{\alpha'}{4\ (x-\alpha')} - \frac{1}{2\ (x-1)} $$

Answer 3

Here it's easier to work with the exponential generating function. You just need to find the Taylor series of each term on the RHS. You already know the Taylor series of one term. To find the Taylor series of the other term, use the fact that $\cos t = \frac{e^{it} + e^{-it}}{2}$.