triangle is a quick and easy way to look up all of these coefficients

You'll see the same thing with n=3,which expands to this.So the triangle is a quick and easy way to look up all of these coefficients.But there's much more.For example, add up the numbers in each row, and you'll get successive powers of two.Or in a given row, treat each number as part of a decimal expansion.In other words, row two is (1x1) + (2x10) + (1x100).
You get 121, which is 11^2.