Given a binomial distribution with parameters $ n $ and $ p $, where $ n $ is an odd integer greater than or equal to 3, I am interested in the truncated binomial distribution where we truncate at $ k = \frac{n+1}{2} $ (i.e., $ X \geq k $). I would like to express the expectation and variance of this truncated distribution using only $ n $ and $ p $, and without involving summation or complicated series expressions.

My ultimate goal is to determine the sign of the following expression: $$ (E - np)(2p - 1) + D - np(1 - p) + (E - np)^2 $$ where $ E $ is the expectation and $ D $ is the variance of the truncated distribution.

Is there a way to express the expectation and variance of this truncated distribution in terms of $ n $ and $ p $ without using summation symbols or recursive relationships? I am open to using standard special functions, but would prefer to avoid summation signs if possible.