You weren't paying attention all those years, were you?
Sinatra recorded exactly one album with Basie and his band.
Correction: Frank recorded 2 studio albums with William Basie..the first album “Sinatra-Basie” was released in January of 1963; the second studio album “Sinatra-Basie-It Might As Well Be Swing” was released in August of 1964, then..they recorded a LIVE album together “Sinatra At The Sands”, which was arranged and conducted by Quincy Jones, released in August of 1966. Soo..yes.. Frank loved to sing with Basie’s band..they even did a tv special together in the late 70’s.