Rich young stellar clusters produce H ii regions whose expansion into the nearby molecular cloud is thought to trigger the formation of new stars. However, the importance of this mode of star formation is uncertain. This investigation seeks to quantify triggered star formation (TSF) in IC 1396A (aka the Elephant Trunk Nebula), a bright-rimmed cloud (BRC) on the periphery of the nearby giant H ii region IC 1396 produced by the Trumpler 37 cluster. X-ray selection of young stars from Chandra X-ray Observatory data is combined with existing optical and infrared surveys to give a more complete census of the TSF population. Over 250 young stars in and around IC 1396A are identified; this doubles the previously known population. A spatio-temporal gradient of stars from the IC 1396A cloud towards the primary ionizing star HD 206267 is found. We argue that the TSF mechanism in IC 1396A is a radiation-driven implosion process persisting over several million years. Analysis of the X-ray luminosity and initial mass functions indicates that >140 stars down to 0.1 M⊙ were formed by TSF. Considering other BRCs in the IC 1396 H ii region, we estimate the TSF contribution for the entire H ii region exceeds 14–25 per cent today, and may be higher over the lifetime of the H ii region. Such triggering on the periphery of H ii regions may be a significant mode of star formation in the Galaxy.