Suppose we have an ordered list of real values: $$x_1,x_2,x_3,...,x_N \in \mathbb R$$
Now if we consider the set in $\mathbb R^{2}$ (ie with their ordering index as the first variable) :
$${\{(1,x_1),(2,x_2),(3,x_3),...,(n,x_n),...,(N,x_N)}\}$$
I have found that the formula used by the computer algebra system I use to produce a polynomial that serves as an interpolation for this set of data points must be:
$$p_N(n)=\left( -1 \right) ^{N}\sum _{k=0}^{N}{\frac {n \left( n-1 \right) !\,\left( -1 \right) ^{k}x_{{k}}}{ \left( n-N-1 \right) !\, \left( n-k\right) k!\, \left( N-k \right) !}}$$
For example a set of 3 data points: $${\{(0,x_0),(1,x_1),(2,x_2)}\}$$
would produce an interpolation of: $$p_2(n)=\left( \frac{1}{2}\,x_{{2}}-x_{{1}}+\frac{1}{2}\,x_{{0}} \right) {n}^{2}+ \left( -\frac{1}{2} \,x_{{2}}+2\,x_{{1}}-\frac{3}{2}\,x_{{0}} \right) n+x_{{0}}$$
So my question regarding this, is the following:
1) What would be the best way of writing the out the requisites for $p_N(n)$ to be assured as having a specific number of integer coefficients?
Additional non specific question:
I feel that this statistical method is not at all helpful (well, as far as the mathematical understanding is concerned as or furthering my insight into the data I am interpolating)and I was wondering if any methods of deduction for "likely to be accurate extrapolation" functions for an infinite subset of $\mathbb R^{2}$ could be possible?
Or put another way, what are the first characteristics I should consider if I were to observe a set of the polynomials that are produced in the aforementioned manner as I increase the cardinality of the subset ($N$), my firsts thoughts where to ask myself if these polynomials are odd or even functions, (which would effectively half the number of functions of consideration for a form that would be more reliable than a polynomial if it were empirical data I was attempting to model) but is there any other properties that are equally as general? Even a good book reference on this subject will get the upvote here really.