You can now perform leave-one-out meta-analysis by using the new option leaveoneout with meta summarize and meta forestplot.
The leave-one-out meta-analysis performs multiple meta-analyses by excluding one study at each analysis. It is common for studies to produce exaggerated effect sizes, which may distort the overall results. The leave-one-out meta-analysis is useful to investigate the influence of each study on the overall effect-size estimate and to identify influential studies. See [META] meta summarize and [META] meta forestplot.