Let $r,w \in \mathbb{N}$. Are there some formula for the next sum?
$${r \choose 1}+{r \choose 2}\cdots{r \choose w}$$ where $w<r$?
2026-04-11 13:05:57.1775912757
Closed formula for ${r \choose 1}+{r \choose 2}\cdots{r \choose w}$ where $w < r$
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There is no closed form for this. See this MO question. One answer rewrites the sum in terms of the hypergeometric function ${}_2F_1$.