Alternating Inverse-Binomial Double Series

We study inverse-binomial double series involving ordinary and alternating harmonic numbers. Using iterated integrals associated with admissible 3-posets, we establish twenty-four explicit evaluations in terms of multiple zeta values and alternating multiple zeta values. The resulting identities involve modified Bell polynomials, generalized harmonic numbers, alternating harmonic numbers, and the factors (−1)m and 2−n. Our approach decomposes each iterated integral into two suitable factors, expands these factors into power series, and evaluates the remaining integral by the beta integral, leading naturally to inverse-binomial coefficients. This yields a systematic framework for evaluating a broad class of inverse-binomial double series. The twenty-four identities further provide building blocks for evaluating more general inverse-binomial double series, as illustrated by several examples.