2 dimensional non-linear recurrence $f(n,k) = \frac{f(n-1,k) + f(n-1,k-1)}{2} - \frac{\left(f(n-1,k) - f(n-1,k-1)\right)^2}{320}$

Question

$f(n,k) = \frac{f(n-1,k) + f(n-1,k-1)}{2} - \frac{\left(f(n-1,k) - f(n-1,k-1)\right)^2}{320}$,

with boundary conditions

$f(n,k) = 0$ $\;$ if $\;$ $n \leq k$,

$f(n,k) = 20n$ $\;$ if $\;$ $k = 0$.

While this is easy to solve programatically using recursion and memoization, I wonder

1. Is there a solution for this recurrence?
2. If not, is there an exact or approximate solution in the limit when $n\to \infty$ and $\frac{k}{n} \to \gamma \in [0,1]$?