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Consider a beta binomial distribution where the number of trials, $n$, is odd and the shape parameters of the underlying beta distribution, $\alpha$ and $\beta$, are equal.

Is there a closed form solution for the expected number of successes conditional on at least half of the trials being successes?

In other words, is there a closed form for \begin{align} 2\sum_{k=\frac{n+1}{2}}^{n} k {n\choose k} \frac{B(k + \alpha, n-k + \alpha)}{B(\alpha,\alpha)}? \end{align}

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