I am looking for a tight upper bound for the following sum:
$$\sum_{m=1}^k \binom{n}{m} \binom{k-1}{m-1} $$
with $k \leq n$ and $m \leq n$.
2026-04-02 17:29:03.1775150943
Tight bound for a sum of binomial coefficients
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$$\sum_{m=1}^k{n\choose m}{k-1\choose m-1}=\sum_{m\in\mathbb{Z}}{n\choose m}{k-1\choose m-1}=\sum_{m\in\mathbb{Z}}{n\choose m}{k-1\choose k-m}={n+k-1\choose k}$$