What is this tune from The Lucy Show and Sooper Goop?

In Season 5 Episode 3 of The Lucy Show, “Lucy the Bean Queen” (1966), Ed Begley (Sr.) sings a jingle about a fictitious brand of beans. You can listen to it at the 6:15 mark, here: https://www.youtube.com/watch?v=_36_dTq_vJQ

The same tune is used in the animated film Sooper Goop (1976) as a jingle for a fictitious brand of cereal. You can hear it at the 1:10 mark, here (make sure that Dailymotion is not muted): https://www.dailymotion.com/video/xem0tb

Can someone please identify this tune? It sounds like some kind of children’s song/nursery rhyme, but I can’t quite place it.

My Pony Boy, a hit from the first decade of last century. (Usually called just “Pony Boy” now.)

Solved! By Jove, that was fast!

I had never heard the original title or lyrics, only the tune. (I have, however, heard other songs sung by Ada Jones.)

Thanks, Hatchie!

Also used by Pat Paulsen for his “Hobby Hot” jingle (on the Smothers Brothers show).

“Hobby hut, hobby hut, you can be a hobby nut.”… etc.