Basically the title. There's three schools of thought.
A) The Harder They Fall would not trigger when you use Improved Knockdown, only regular Knockdown, because the Improved Knockdown feat only applies the critical success effect of a trip. This doesn't mean you "critically succeed at tripping a foe", only that you get to apply the "critical success" effect under the "trip" action to your strike.
B) The Harder They Fall would trigger when you use Improved Knockdown, as the effects of The Harder They Fall are now part of the "critical success effect of a trip".
C) The Harder They Fall would trigger when you use Improved Knockdown, because any action which applies the success or critical success effects of an action counts as succeeding or critically succeeding at that action for the purpose of feats related to that action, respectively.