At 0:40 of the documentary A Great Day in Harlem, we hear "One O'Clock Jump" by Count Basie Orchestra:
What year is this specific version of "One O'Clock Jump"?
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I believe this version is from 1957, a live recording Count Basie did at the Newport Jazz Festival with his Orchestra. It can be found on Youtube here. According to Wikipedia, Basie's At Newport album, on which this version appears, was recorded on July 7, 1957, and the album was released in early November of the same year.
Fun fact about this recording: according to all of these different sources, the original recording Count Basie's "One O'Clock Jump" was done on July 7, 1937, meaning that this historic performance was the 20th anniversary of the original recording.