The answer is MacMahon himself, who proved this in his book Combinatory Analysis as a corollary of a more general theorem about plane partitions. See Sections IX and X.

The answer is MacMahon himself, who proved this in his book Combinatory Analysis as a corollary of a more general theorem about plane partitions. See Sections IX and X.

There is some additional historical information in the Notes to Chapter 7 of Richard Stanley's book Enumerative Combinatorics, volume 2.