According to Allmusic.com, several different people have written songs called INVITATION TO THE BLUES:

One written by Doris Fisher, Arthur Gershwin & Allan Roberts, recorded by:
Les Brown & His Orchestra
Doris Day
Anita Ellis
Lannie Garrett
Gene Krupa & His Orchestra
Julie London
Ella Mae Morse

One written by Mike Titre & Todorovic, recorded by:
Blues Company

One by Roger Miller, recorded by:
Tommy Alverson
Pete Drake & His Talking Steel Guitar
Bill & Bonnie Hearne
Chris Hillman
Jim & Jesse
Kiss
Reba McEntire
Roger Miller
Weldon Myrick
Willie Nelson
Ray Price and the Cherokee Cowboys
Doug Sahm
Jumpin' Gene Simmons
Wild River Band
Undertakin' Daddies

One written by Tom Waits, recorded by:
David Basse
Holly Cole Trio
James Naughton
Giant Sand
Bill Sheffield
Tom Waits
Jennifer Warnes

D. W. Ditty's recording is also listed, but it doesn't say who wrote it, and there is no sound sample.