Who Wrote the Poem About the Purple Cow?

Ages ago I was at my grandmother's house, reading a book of poetry by Ogden Nash, with illustrations. The poem that I remember the clearest, and we're talking about a book I read nearly 30 years ago in seventh grade, was the one about the purple cow. You probably know the one, "I never saw a purple cow, And never hope to see one" etc.

Today I was looking at an entry at Merriam-Webster's website and they say that a fellow called Gelett Burgess wrote it.

If you do a google you'll quite a lot of hits saying Ogden Nash wrote the poem and the followup regretting writing the poem and a couple saying no, Burgess really wrote it.

I'm sure the book I read said Nash wrote it.

Who did and how did it all get so tangled up in the memory of those who read it?