The sense from “Feel Good Blues” is that this was the Count Basie Band just having a whole of fun. There are smiles all around and Basie talks over the music a couple of times. The Count samples at least two standards (“Blues in the Night” and “Salt Peanuts”) at the start. There may be others that I don’t recognize.
Nothing sounds quite as good and distinctive as The Count Basie Orchestra. The lineup, according to the video: Freddie Green (guitar), Cleveland Eaton II (bass), Gregg Field (drums), Jo Jones (guest, drums), Sonny Cohen, Peter MInger, Darle Carley and Frank Szabo on trumpets, Bill Hughes, Dennis Wilson, Grover Mitchell and Booty Wood (trombones), Bobby Platter (saxophone), Kenny Hing and Eric Dixon (tenor sax), Danny Turner (alto sax) and Johnny Williams Jr. (baritone sax).
Nobody was cooler than Count Basie. Check him out here and here as well.
Here is “The Count Meets the Duke” — i.e., Msesrs. Basie and Ellington — and Basie’s autobiography, “Good Morning Blues,” at Amazon. The Reprise recordings of Frank Sinatra and Basie is available at iTunes.