They must have known about Rothbard's book, because they quote a paper by Edward Glaeser twice, and both quotes come from a paragraph that mentions or cites Rothbard's seminal work. They also refer to a blog by Walter Coffey, who cites Rothbard's Panic of 1819 at the bottom.
Rothbard's book is considered the standard on the subject of the 1819 crisis, yet the NY Fed authors seem reluctant to speak his name.
Consider these similarities (I've underlined some particularly similar wording):
NY Fed article: "In the mid-1810s, the nation was still sparsely populated at just 7 million inhabitants, the economy was agriculture-based, specie was scarce, andbarter was extensive on the frontier."
Rothbard, p. 1: "Before the war, America had been a large, thinly populated country of seven million, devoted almost exclusively to agriculture. Much cotton, wheat, and tobacco were exported abroad, while the remainder of the agricultural produce was largely consumed by self-sufficient rural households. Barter was extensive in the vast regions of the frontier."
NY Fed article: "It was during this period that future President Andrew Jackson shaped his anti-Bank views in Tennessee while his future hard-money arm in the Senate, Thomas Hart Benton (Old Bullion), shaped his views in Missouri, two of the hardest-hit states."
Rothbard, p. 248: "Andrew Jackson himself foreshadowed his later opposition to banking by making himself the fervent leader of the opposition to inconvertible paper in Tennessee. Thomas Hart (“Old Bullion”) Benton, later Jackson’s hard money arm in the Senate, was converted to hard money by his experience with banking in Missouri during the panic."
NY Fed article: "The nation was leery of a national bank with seemingly endless power to manipulate the money supply and the Second National Bank of the United States was attacked by both the expansionists and the sound money opponents."
Rothbard, p. 244: "It is often overlooked, however, that hostility to the bank on economic grounds came from two opposing directions: from those who attacked it as too restrictive, and from the hard money ultras who considered it a nationwide engine of monetary expansion."
These two groups were called the "expansionists" and "sound money opponents" in the same section of text.