Type 2, No Drapery

Ron Guth: The first modification to the Liberty Seated half dime design took place in 1838, when 13 stars were added to the obverse in a large arc around Miss Liberty following an order from Mint Director Robert Maskell Patterson. Until 1840, the "With Stars" half dimes were created using a hub of the No Stars design, followed by the addition of the stars by hand. This type is differentiated from later issues by the lack of drapery underneath Liberty's arm (holding the pole). The 1838 Small Stars resulted from overzealous polishing of the die, thereby reducing the size of the stars (unlike the 1838 Small Stars dime, where a star punch from half dimes was used). The 1840-O Large Letters Reverse is a rare transitional piece, combining a No Drapery obverse with a later reverse.

Sources and/or recommended reading:
"Walter Breen's Complete Encyclopedia of U.S. and Colonial Coins" by Walter Breen