How to obtain probability distribution from the generating function $G(s) = e^{a(s-1)^2}$? I was trying to get the probability distribution $p(n)$ from a generating function $G(s)$ like this: 
$G(s) = e^{a(s-1)^2}=\sum s^np(n)$
I need first to do Maclaurin expansion of the exponential and then get the $n$th order term for $p(n)$.
My first thinking was it would be simple to calculate the derivatives. But it turns out to be much more difficult and also very interesting to generalize the $n$ order derivative. 
I list a table for the powers and coefficients of each derivative order, finding that the powers are odd numbers for odd $n$, even numbers for even $n$, the coefficients are associated to $(n-1)$th order's powers and coefficients. It is easy to see the association but I cannot generalize it. 
This calculation is also like $z$ transform but no existing result for it. 
Anyone could give a shot and help me out?
 A: Using Maple and  the information given at
http://en.wikipedia.org/wiki/Hermite_distribution
it is possible to obtain 

The first ten probabilities are

These results can be obtained as follows.
The generating function for the Hermite polynomials is
$${{\rm e}^{2\,xt-{t}^{2}}}=\sum _{n=0}^{\infty }{\frac {H_{{n}} \left( 
x \right) {t}^{n}}{n!}}$$
Then 
$${{\rm e}^{-\alpha\, \left( 1-s \right) ^{2}}}={{\rm e}^{-\alpha}}{
{\rm e}^{2\,s\alpha-\alpha\,{s}^{2}}}
$$
making the change $$t=\sqrt {\alpha}s$$ we have
$${{\rm e}^{-\alpha\, \left( 1-s \right) ^{2}}}={{\rm e}^{-\alpha}}{
{\rm e}^{2\,t\sqrt {\alpha}-{t}^{2}}}
$$
and then we obtain
$${{\rm e}^{-\alpha}}{{\rm e}^{2\,t\sqrt {\alpha}-{t}^{2}}}={{\rm e}^{-
\alpha}}\sum _{n=0}^{\infty }{\frac {H_{{n}} \left( \sqrt {\alpha}
 \right) {t}^{n}}{n!}}$$
it is to say
$${{\rm e}^{-\alpha\, \left( 1-s \right) ^{2}}}={{\rm e}^{-\alpha}}\sum 
_{n=0}^{\infty }{\frac {H_{{n}} \left( \sqrt {\alpha} \right)  \left( 
\sqrt {\alpha}s \right) ^{n}}{n!}}$$
Then we write
$$ \sum _{n=0}^{\infty }p_{{n}}{s}^{n}={{\rm e}^{-\alpha}}\sum _{n=0}^{
\infty }{\frac {H_{{n}} \left( \sqrt {\alpha} \right)  \left( \sqrt {
\alpha}s \right) ^{n}}{n!}}$$
Finally we obtain
$$p_{{n}}={\frac {{{\rm e}^{-\alpha}}H_{{n}} \left( \sqrt {\alpha}
 \right)  \left( \sqrt {\alpha} \right) ^{n}}{n!}}$$
A: $G$, using your notation, is known in probability as the probability generating function. See here for more details. It is seen on this link that 
$$p(k) = \dfrac{\left.G^{(k)}(s)\right|_{s=0}}{k!}$$
where $G^{(k)}$ denotes the $k$th derivative.
