This unusual variety shows 15 stars on the obverse instead of the usual 13. This was strictly an engraving error that occurred when the stars were placed too closely together. It is in no way related to the earliest U.S. coins that had varying number of stars. In those cases, the number of stars started at 13, then increased with the addition of each new state, but the number eventually reverted back to 13 as the designs became too crowded. By 1817, there were 18 states in the Union, thus there is no valid connection with the number 15.
The 1817 15 Stars Large Cent is a common variety which can be found with relative ease in circulated grades. Mint State examples are quite scarce (as are most Large Cents of this year). According to researcher John Wright, there are numerous Mint State examples of this variety, but PCGS has graded barely ten in the last 25+ years. The finest example certified by PCGS is a single MS65BN. Red & Brown examples are extremely rare and no full red example is known.