De Moivre gave a simple closed form expression for the mean absolute deviation of the binomial distribution. Later authors showed that similar closed form expressions hold for many of the other classical families. We review the history of these identities and extend them to obtain summation formulas for the expectations of all polynomials orthogonal to the constants.
"Closed Form Summation for Classical Distributions: Variations on a Theme of De Moivre." Statist. Sci. 6 (3) 284 - 302, August, 1991. https://doi.org/10.1214/ss/1177011699