Rosemary Clooney recorded a song called "Shaun, Shaun, the Leprechaun" (sic) in 1954. (I suppose Americans in those days didn't know how to spell "Sean".) It is included on her 7-CD compilation set "Memories of You", Bear Family CD 15914, 1998. I transcribed this from a sound sample:

Oh, Sean, Sean, the leprechaun lives in big old tree,
And all the children look for him, toora-loora-lee.
So look for Sean the leprechaun and do just what you're told,
And, toora-loora-loora, he'll give you a pot of gold.

He can make it rain by just blowing a note upon his silver horn.
The sun won't shine till you fill up his hat with a ...

Rolf Harris recorded a song (and an album) called "Shamus O'Sean, the Leprechaun".

A 1972 LP titled "Golden Hour Of Children's TV Themes" contains a song called "Shaun the Leprechaun".

This little rhyme (with various spellings) was found on several pages of children's rhymes:

SEAN, SEAN, THE LEPRECHAUN

Sean, Sean, the leprechaun
Danced through the forest with a green suit on.
You can chase him all day long.
Come too close, and poof! he's gone.