The Lonnie Donegan version which we Brits learned in 1956-ish implied Pakenham was on the other side:

"In 1814 we took a little trip,
Along of Col Pakenham down the mighty Mississip
We brough a little bacon and we brought a little beans
And we fought the blooming British at a town called New Orleans"

Although the error was quickly pointed out by historians, I don't think his version ever changed.
RtS