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 $w<r$?
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http://math.stackexchange.com/questions/734900/proof-by-induction-sum-of-binomial-coefficients-sum-k-0n-n-k-2n , I think this is your question except they begin what you are calling w at. Since the $0th$ zero of your sum would be 1 your sum is $2^r-1$. – mike van der naald Jul 20 '16 at 01:08
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@mikevandernaald thanks, by in my case $w<r$ – juaninf Jul 20 '16 at 01:11
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1No, doesnt exist closed formula for partial sums of binomial coefficients when the variable is $w$. But exist closed formula for some other cases, by example when the variable is $r$. – Masacroso Jul 20 '16 at 01:13
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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 ${}_2F_1$.
