The First Bank of the United States and the STate Banks

I was reading Rothbard’s History of Money and Banking in the United States, and Murray Rothbard says that the First Bank of the United States aided in inflationism. Though this seems to be true judging from the state banks’ support for the First Bank and both the state banks’ and the First Bank’s fractional reserve, how specifically did the First Bank of the United States cooperate with the state banks to this end?