"I'm Gonna Sit Right Down and Write Myself a Letter" is a 1935 popular song with music by Fred E. Ahlert and lyrics by Joe Young. It has been recorded many times, and has become a standard of the Great American Songbook. The artists recording the song include Frank Sinatra on his 1954 album Swing Easy! and Bing Crosby on his 1956 LP Bing with a Beat.
the Fats Waller version on Sesame Street
But the first big hit version was by Fats Waller in 1935; the song also had a major revival in 1957 in a recording by Billy Williams, which reached #3 on the Billboard magazine charts. Among other versions are recordings by Barry Manilow, Dean Martin, Scatman Crothers, and Bill Haley & His Comets, the latter who recorded a rock and roll version of the song in 1957.
Dean Martin's interpretation, used as an opening number for his show: