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In this paper, we introduce and study new classes of Ap\'ery-type seriesinvolving the multiple $t$-harmonic sums by combining the methods of iteratedintegral and Fourier--Legendre series expansions, where the multiple$t$-harmonic sums are a variation of multiple harmonic sums in which all thesummation indices are restricted to odd numbers only. Our approach also enablesus to generalize some old classes of Ap\'ery-type series involving harmonicsums to those with products of harmonic sums and multiple $t$-harmonic sums. Inall of these series, the central binomial coefficients appear as $a_n^{\pm 1}$or $a_n^{\pm 2}$ where $a_n=\binom{2n}{n}/4^n$. We show that every such seriescan be expressed as either the real or the imaginary part of a $\mathbbQ$-linear combination of colored multiple zeta values of level 4.

Ap\'ery-Type Series and Colored Multiple Zeta Values