This was on HN a couple of days ago:

  http://www.oftenpaper.net/sierpinski.htm

It points out that you can interpret the table as the relation "is disjoint" between the bitsets encoded in the row and col indices (CTRL+F for "binary logic"). Following links from there will take you to a rabbit hole of combinatorics that I will not claim to fully understand.

The whole page is a delight, though, and given your enjoyment for your discovery, you'll certainly love browsing it.