The actual name of this song is “Linus and Lucy.” It was recorded in 1964 by Vince Guaraldi. This piece was commissioned by Lee Mendelson for the upcoming movie, “A Boy Named Charlie Brown.”

From Wikipedia, “Within several weeks, Mendelson received a call from an excited Guaraldi who wanted to play a piece of music he had just written. Mendelson, not wanting his first exposure to the new music to be marred by the poor audio qualities of a telephone, suggested coming over to Guaraldi’s studio. Guaraldi enthusiastically refused, saying ‘I’ve got to play this for someone right now or I’ll explode!’ He then began playing the then untitled “Linus and Lucy” for Mendelson, who agreed the song was perfect.”