No your eyes aren’t deceiving you. The American flag that hangs in Ashokan’s 1817 Schoolhouse has 15 stars AND 15 stripes.
Here is a little history of the flag:
The 15-star, 15-stripe flag was the official United States flag from 1795 to 1818, and was the only US flag to have more than 13 stripes. The flag was authorized by Congress on January 13, 1794, and went into effect on May 1, 1795. The additional stars and stripes represented the admission of Vermont in 1791 and Kentucky in 1792 to the Union, as it was customary to add one star and stripe for each new state.
The original flag was sewn by prominent Baltimorean flagmaker Mary Young Pickersgill under a government commission in 1813 at a cost of $405.90 (equivalent to $6,408 in 2023). Armistead specified “a flag so large that the British would have no difficulty seeing it from a distance”.
The flag that flew over Fort McHenry during the British attack in 1814 inspired Francis Scott Key to write the national anthem, The Star-Spangled Banner. The flag that flew over the fort on that day is now preserved in the Smithsonian Museum.
In 1795, the number of stars and stripes was increased from 13 to 15 (to reflect the entry of Vermont and Kentucky as states of the Union). For a time the flag was not changed when subsequent states were admitted, probably because it was thought that this would cause too much clutter. It was the 15-star, 15-stripe flag that inspired Francis Scott Key to write “Defence of Fort M’Henry”, later known as “The Star-Spangled Banner”, which is now the American national anthem. The flag is currently on display in the exhibition “The Star-Spangled Banner: The Flag That Inspired the National Anthem” at the Smithsonian Institution National Museum of American History in a two-story display chamber that protects the flag while it is on view.
On April 4, 1818, a plan was passed by Congress at the suggestion of U.S. Naval Captain Samuel C. Reid in which the flag was changed to have 20 stars, with a new star to be added when each new state was admitted, but the number of stripes would be reduced to 13 so as to honor the original colonies. The act specified that new flag designs should become official on the first July 4 (Independence Day) following the admission of one or more new states.