Rotation Billiards

A professional pocket billiard player practices every day by playing a kind of rotation game. The balls are numbered from 1 to 15, and on the wall hangs a rack with 15 numbered cubicles. As each ball is pocketed, it is removed from the pocket and placed in the cubicle of the same number. Just to make it harder, the player stipulates that the row of balls in the rack must never at any time show an interior gap. Thus, if the 6 is pocketed first, the 5 or the 7 must be pocketed next. There is free choice of which ball to sink first, but after that the choice is restricted to balls numbered in sequence with those already in the rack.

The question is, in how many different orders can the player clean the table of all 15 balls?