The moments of the unboounded metalog distribution can be all expressed as a closed-form function of the scaling constants (a1, ... an). Generally, the kth central moment of the n term metalog is a kth-order polynomial of the ai's. See The Metalog Distributions, Section 3.4, for further explanation. You may download these moments below.
To view the first four central moments (mean, variance, skewness, and kurtosis) for unbounded metalog distributions of up to ten terms, click the icon on the right. For the Excel implementation of these moments, download the "Unbounded Metalog" Excel workbook.