Probability that a clumsy boy eats $k$ out of 20 candies

Setup on recursion:

$p\left(n,i\right)$ probability that $i$ of $n$ candies are eaten.

To be found: $p\left(20,k\right)$.

$p\left(n,i\right)=0$ if $2i<n\vee i>n$

$p\left(0,0\right)=1=p\left(1,1\right)$

If $2i\geq n\geq2$ then: $$p\left(n,i\right)=0.04\left(n-1\right)p\left(n-2,i-1\right)+\left[1-0.04\left(n-1\right)\right]p\left(n-1,i-1\right)$$

I have an excel sheet with the outcomes for $n\leq20$.

Looking at the highest outcome: the probability that the boy will eat $15$ of $20$ candies is about $0.329869$.