In my research, I obtained a sequence of polynomials (I am only able to compute the first 4 of them): \begin{align} & f(2) = 1+t, \\ & f(3) = 1+4t+3t^2, \\ & f(4) = 1+6t+12t^2+7t^3, \\ & f(5) = 1+8t+20t^2+28t^3+15t^4. \end{align} Is it possible to find the general formula of $f(n)$ using these 4 polynomials? Some patterns are:

- The constant term is 1.
- The highest degree term is $(2^{n-1}-1)t^{n-1}$.
- $(1+t)$ is a factor of each polynomial.

Thank you very much.