Based on the original sizes you gave, it could've been a Modern Jazz Outfit like I mentioned earlier. With the corrected 20-13-16 sizes you listed, it could've been an Ensemble 1-N Gene Krupa kit. But whether or not it was a cataloged configuration, it really is "some old Slingerland", circa 1968-69 or so.
Drum manufacturers back then didn't have different models with different shell configurations like they do nowadays. I mean, today you can buy drums from a company in ash, maple, birch, 6-ply, 8-ply, etc. You can buy Tama Imperialstars, Rockstars, Superstars, Swingstars, etc. But back then, if you bought Slingerland drums, they were 3-ply maple-poplar-mahogany (or maple-poplar-maple for drums with painted/lacquered finishes.) It didn't matter if you bought the top of the line set in the catalog or the most basic drums. Same drums, just sold in different combinations and with different options. And for the most part, that was how the other major drum companies operated, too.
Also, when a drum shop or customer ordered new drums, it was common to deviate from the cataloged sets. Even on the day it was brand new, your drums might not be an exact match for a cataloged kit.
One last thing... I think your sizes are wrong. I'm guessing you measured the depths with the heads on, and maybe only measured the space between the hoops? (There's a little bit more shell left under those heads & hoops.)
Likely sizes, depth x diameter:
Bass = 14" x 20"
Mounted tom = 9" x 13"
Floor Tom = 16" x 16"
Snare = 5.5" x 14"