First, one of the nicest and most genuine men to have ever walked the face of this imperfect planet, Fred Rogers (I cried like a baby the day he died, as did you). Fred Rogers was of course an accomplished pianist and composer. It seems like he was always taking his viewers on a trip to Negri's Music Shop. Thank you for that, Mr. Rogers!
That theme sure is jazzy!
At about 3:50, and things get really jazzy at around a minute from that.
"The Music of Mr. Rogers' Neighborhood"
Vince Guaraldi and Trio did the music for many of Charles Schulz's "Peanuts" cartoons, with the most obvious perhaps being the wonderful music for A Charlie Brown Christmas (damn I love "Skating"! I could listen to that all year long):
"Mah-Na Mah-Na," originally from Sesame Street(?) and later The Muppet Show:
I would have caught these in rerun form some years after their introduction.
The first season theme music to I Dream of Jeannie was an instrumental jazz waltz written by Richard Wess. From the second season on, however, it was replaced by a new theme entitled "Jeannie," composed by Hugo Montenegro.
The Bewitched theme by Warren Barker