I don't have any answer, but here are a few terms to be sure it is clear and to show some regularity in the constant or linear terms. Writing $f_n := f(n)$ and taking $f_1=1$, it looks like

$$P_0(n)=1$$

$$P_1(n) \overset{?}= \binom{n+1}{1}+\binom{n+1}{2}f_1$$

$$P_2(n) \overset{?}= \binom{n+2}{2}+\binom{n+2}{3}\left(f_2+2f_1\right)+2\binom{n+2}{4}f_1^2 $$

$$P_3(n) \overset{?}= \binom{n+3}{3} +\binom{n+3}{4}\left(f_3 +2f_2 +3f_1\right) +5\binom{n +3}{5}\left(f_2f_1 +f_1^2\right) +5\binom{n +3}{6}f_1^3$$

$$P_4(n) \overset{?}= \binom{n+4}{4} +\binom{n +4}{5}\left(f_4 +2f_3 +3f_2 +4f_1\right) +3\binom{n +4}{6}\left(f_2^2 +2f_3f_1 +3f_1^2 +4f_2f1\right) +7\binom{n +4}{7}\left(2f_1^3+3f_2f1^2\right) +14\binom{n +4}{8}f_1^4$$

$$P_5(n) \overset{?}= \binom{n+5}{5} +\binom{n +5}{6}\left(f_5 +2f_4 +3f_3 +4f_2 +5f_1\right) +7\binom{n +5}{7}\left(f_3f_2+f_2^2 +f_4f_1 +2f_3f_1 +2f_1^2+3f_2f_1\right) +28\binom{n +5}{8}\left(f_2^2f_1 +f_3f_1^2 +f_1^3 +2f_2f_1^2\right) +42\binom{n +5}{9}\left(f_1^4+2f_2f_1^3\right) +42\binom{n +5}{10}f_1^5$$

I don't have any answer, but here are a few terms to be sure it is clear and to show some regularity in the constant or linear terms. Writing $f_n := f(n)$ and taking $f_0=1$, it looks like

$$\begin{array}{r,l}P_0(n)=&1\\ P_1(n) \overset{?}=& \binom{n+1}{1}+\binom{n+1}{2}\;\;f_1\\ P_2(n) \overset{?}=& \binom{n+2}{2}+\binom{n+2}{3}\left(f_2+2f_1\right) &+&\dots\\ P_3(n) \overset{?}=& \binom{n+3}{3} +\binom{n+3}{4}\left(f_3 +2f_2 +3f_1\right) &+&\dots\\ P_4(n) \overset{?}=& \binom{n+4}{4} +\binom{n +4}{5}\left(f_4 +2f_3 +3f_2 +4f_1\right) &+&\dots\\ P_5(n) \overset{?}=& \binom{n+5}{5} +\binom{n +5}{6}\left(f_5 +2f_4 +3f_3 +4f_2 +5f_1\right) &+&\dots \end{array}$$ $$\begin{array}{r,l} P_2(n) \overset{?}=&\dots&+&2\binom{n+2}{4}\;\;f_1^2\\ P_3(n) \overset{?}=&\dots&+&5\binom{n +3}{5}\left(f_2f_1 +f_1^2\right) &+&\dots\\ P_4(n) \overset{?}=&\dots&+&3\binom{n +4}{6}\left(f_2^2 +2f_3f_1 +3f_1^2 +4f_2f1\right) &+&\dots\\ P_5(n) \overset{?}=&\dots&+&7\binom{n +5}{7}\left(f_3f_2+f_2^2 +f_4f_1 +2f_3f_1 +2f_1^2+3f_2f_1\right) &+&\dots \end{array}$$ $$\begin{array}{r,l} P_3(n) \overset{?}=&\dots&+&\;\;5\binom{n +3}{6}\;\;f_1^3\\ P_4(n) \overset{?}=&\dots&+&\;\;7\binom{n +4}{7}\left(2f_1^3+3f_2f1^2\right) &+&\dots\\ P_5(n) \overset{?}=&\dots&+&28\binom{n +5}{8}\left(f_2^2f_1 +f_3f_1^2 +f_1^3 +2f_2f_1^2\right) &+&\dots \end{array}$$ $$\begin{array}{r,l} P_4(n) \overset{?}=&\dots&+&14\binom{n +4}{8}\;\;f_1^4\\ P_5(n) \overset{?}=&\dots&+&42\binom{n +5}{9}\left(f_1^4+2f_2f_1^3\right) &+&42\binom{n +5}{10}\;\;f_1^5 \end{array}$$