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Estimating when does a certain binomial sum exceed an upper bound

Given a fixed integer $n > 0$ and $0 \le m \le n$ let us define the numbers $$f_{n,m} = \sum_{i=\lfloor m/2 \rfloor}^m {n-2i \choose n - m -i}{i+1 \choose m - i +1}.$$ For example $f_{n,0} = ...