Grateful Dead or Allman Brothers: Who was the original "jam" band?

Just recently I heard a recording by Phil Lesh (formerly of the Greatful Dead) & Friends Instant Live. Although I have listened to a lot of “Dead” and derivatives over the years, I was struck by the similarity in sound to the Allman Brothers Band. They were both extremely popular in the late sixties and early seventies, (and beyond), and both were known for long improvisations or jams.

I guess the question comes down to this: Were Jerry Garcia and the Dead influenced by Duane and Gregg Allman, et al, or vice versa, or were both groups influenced by the so-called “psychedelic” jams that were popular with so many groups of the time?