hypothesis: for any finite sequence of random positive integers exists a generating function.
If the hypothesis is true, is there a structured way of finding a generating function for a finite sequence of random positive integers?
Here is what I understand from the past 2 days of reading:
A generating function for Fibonacci numbers - eventhough it doesn't have much to do with my question - is: \begin{eqnarray*} \frac{1}{1− (x + x ^ 2)} \end{eqnarray*} where: \begin{eqnarray*} \frac{1}{1− (x + x ^ 2)} =1 + x + 2 x ^ 2 + 3 x ^ 3 + 5 x ^ 4 + 8 x ^ 5+ \cdots \end{eqnarray*} The coefficients are Fibonacci numbers, i.e. the sequence {1,1,2,3,5,8,13,21, ...}.
But if I have a finite sequence of random numbers, for example {4,5,0,9,1,5} Is there a method of finding its generating function?