Vtyjkyuk

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:

$$p_N(n)=left( -1 right) ^Nsum _k=0^Nfrac n left( n-1 right) !,left( -1 right) ^kx_k left( n-N-1 right) !, left( n-kright) k!, left( N-k right) !$$

For example a set of 3 data points:
$$(0,x_0),(1,x_1),(2,x_2)$$

would produce an interpolation of:
$$p_2(n)=left( frac12,x_2-x_1+frac12,x_0 right) n^2+ left( -frac12
,x_2+2,x_1-frac32,x_0 right) n+x_0$$

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.