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Theoretical predictions suggest that very massive stars have the potential toform through multiple collisions and eventually evolve into intermediate-massblack holes (IMBHs) within Population III star clusters that are embedded inmini dark matter haloes. In this study, we investigate the long-term evolutionof Population III star clusters, including models with a primordial binaryfraction of $f_{\rm b}=0$ and 1, using the $N$-body simulation code\textsc{petar}. We comprehensively examine the phenomenon of hierarchicaltriple black holes in the clusters, specifically focusing on the merging oftheir inner binary black holes (BBHs), with post-Newtonian correction, by usingthe \textsc{tsunami} code. Our findings suggest a high likelihood of the innerBBHs containing IMBHs with masses on the order of $\mathcal{O}(100)M_{\odot}$,and as a result, their merger rate could be up to $0.1{\rm Gpc}^{-3}{\rmyr}^{-3}$. In the model with $f_{\rm b}=0$, the evolution of these merginginner BBHs is predominantly driven by their gravitational wave radiation at anearly time, but their evolutionary dynamics are dominated by the interactionbetween them and tertiary BHs in the case with $f_{\rm b}=1$. The orbitaleccentricities of some merging inner BBHs oscillate over time periodically,known as the Kozai-Lidov oscillation, due to dynamical perturbations.Furthermore, the merging inner BBHs tend to have highly eccentric orbits at lowfrequency range, some of them with low redshift would be detected byLISA/TianQin. In the higher frequency range, the merging inner BBHs across awider redshift range would be detected by DECIGO/ET/CE/LIGO/KAGRA.

Merging Hierarchical Triple Black Hole Systems with Intermediate-mass Black Holes in Population III Star Clusters