Consider a sequence of numbers $(v_1, \ldots, v_n)$. Now we remove $k$ numbers from the sequence so that we have $k+1$ non-empty segments of numbers. For example, consider $(2,3,8,1,4)$. If we remove 8 then we have two segments $(2,3)$ and $(1,4)$.

Now we want to minimize the maximum sum of numbers of a segment. Let $m(1,n,k)$ be the answer, then what is the correct recursion for $m$ when $1 \le i < j \le n$ and $0 < k \le (j-i)/2$?