Judging by auction appearances, the "Head of 1840" is even more rare than the "Head of 1838" than the above mintages or Coin World Trends would indicate. The demand for the "Head of 1838" is skewed due to type collectors chasing the elusive two-year "Large Letters" type.
The "Head of 1840" variety was first reported by B. Max Mehl in Mehl's Numismatic Monthly in December 1910. He mistakenly believed the variety to be a pattern.
A few proofs are known. These have the "Head of 1838".
A "Head of 1838" variety exists with a repunched date. Breen claims that this is an 1839/8 overdate. I have seen skepticism of this claim in auction catalogs, and it may not deserve a large premium.
A raw MS65 "Head of 1838" variety sold for $253,000 in a Feb. 1998 Superior auction. It was promoted as the finest-known coin of the type.