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:


1. The constant term is 1.
2. The highest degree term is $(2^{n-1}-1)t^{n-1}$.
3. $(1+t)$ is a factor of each polynomial.


Thank you very much.



What are the patterns of the sequence of polynomials?
One way is


$f(n)=(1+t)\left( \sum\_{k=0}^{n-2}(2(2^k-1)(n-k-1)+1)t^k \right)$


