Crossposted from https://math.stackexchange.com/questions/2605895/asymptotics-for-partial-sum-of-product-of-binomial-coefficients For some $ c<n$ and $ 2c\leq x\leq 2n$ , are there references or previous results for determining the asymptotics (as $ n\to\infty$ ) of the partial sum $ $ \sum_{k=x-c}^c\binom{n}{k}\binom{n}{x-k} $ $ or equivalently if $ c=n\lambda_1$ and $ x=2n\lambda_2$ , for constants $ 0<\lambda_2\leq\lambda_1<1$ $ $ \sum_{k=2n\lambda_2-n\lambda_1}^{n\lambda_1}\binom{n}{k}\binom{n}{2n\lambda_2-k} $ $ IRead more