or is it only multiplcation that has a short handy format, and in which case...why?
The reason for this probably comes down to a few factors. Firstly there's no easy formula for factorials. The gamma function does give you factorials, but it's a hideous formula to use. In comparison, 1+2+...+n = n(n
+1)/2, which is both easy enough to remember and very quick to calculate. Secondly, factorials come up a lot in some branches of mathematics, such as combinatorics, analysis and algebra. While triangle numbers do also come up a lot, it's rarely an issue to just use the formulaic version.