For all $n \in \mathbb{N}$, let ${\mathcal A}_n := \left\{\lceil n/2\rceil, \lceil n/2\rceil+1,\dots, n-1 \right\}$ and

$$f(n) := \begin{cases} \min\limits_{a \in {\mathcal A}_n} \frac 1 4 \binom n a f(a) & \text{if $n\geq 4$}\\ \qquad 1 & \text{otherwise} \end{cases} $$

I am looking for a closed-form expression for $f(n)$, but I could not find it.

So far what I have done is to try to get it into Python, I got the first 100 values, and I thought I would be able to guess the closed-form expression, but it is too difficult to guess it.

The first 10 values are:

$$
\begin{array}{cl}
n & f(n) \\
\hline
0 & 1 \\
1 & 1 \\
2 & 1 \\
3 & 1 \\
4 & 1.0 \\
5 & 1.25 \\
6 & 1.875 \\
7 & 3.28125 \\
8 & 6.5625 \\
9 & 14.765625
\end{array}
$$

Putting it in MS Excel I found that it is NOT an exponential expression as I guessed, but I couldn't find any more.