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Let $\gamma>1$ be a real number and let $n\in \mathbb{N}$. Define $f\colon\mathbb{N}\to[0,1]$ $$ f(n_0) = \frac{\binom{n-n_0}{m}}{\binom{n}{m}}, $$ where $$ m = \Big\lfloor{\frac{n}{\lceil\gamma ...

I'm seeking a closed-form expression to the sum $$ \sum_{j=0}^{2m} (-1)^j (m-j)^{2m+2k} \binom{2m}{j} $$ where for positive integers $m$ and $k$, we know $m \gg k$. Loosely, $k \sim \log(m)$. When $k=...