First we make a difference table by subtracting 
each adjacent pair, placing the difference between
and below each pair.  Then doing the same thing
again until the numbers are all the same:

         0     14     52   126   248 
            14     38    74   122  
               24     36    48 
                   12    12

It takes 3 lines before the differences are the same, 12.

Therefore we must assume a 3rd degree polynomial:



When n = 1:





When n = 2:





When n = 3:





When n = 4:



 

 

Do you know how to solve a system of
4 equations in 4 unknowns? If not post again
asking how.

Answer:  , , , 

So  becomes

 or



Edwin