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Sum of squares of middle binomial sums or 'Truncated mean' of binomial coefficients under binomial distribution
$\mu=1+\epsilon$ where $\epsilon>0$ holds.
1.Is there a good bound for $$T=\frac{\sum_{i=-\sqrt{\mu n\ln n}}^{\sqrt{\mu n\ln n}}\binom{n}{\frac n2 +i}^2}{2^n}?$$
This quantity can be ...
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Local behavior of the Vandermonde convolution
An interesting combinatorial identity is the Vandermonde convolution identity:
$$ \sum_k {n\choose k}{m\choose s-k} = {n+m \choose s},$$
which can be proved by considering the coefficients in $(x+1)^{...