Self study question about Padé Approximation and transfer functions
The following might help. Given the data of $5$ samples, 

$$ x[n] = [1, 2, 3, 4, 5]$$

You will have possible choices for the orders $(p,q)$ of $a[k]$ and $b[k]$, such as $\{(4,0),(3,1),(2,2),(1,3),(0,4)\}$. 

You can solve or each of the possibilities and check whether they yield stable systems or not, by looking at the roots of $a[k]$... afaik they should all be causal as this is Pade modeling assumption (unless equations modified to handle otherwise). 

Note that for certain data sets (including this one) some tail coefficients might turn to be $0$ and the actual order can be less than that of indicated by $p$ or $q$.

Also note that a model with one pole is not an all-pole model. All-pole model requires that $q=0$, and does not depend on $p$.