The mathematician Donald Knuth (b. 1938) once indicated that there are so many relations and patterns in Pascal's triangle that when someone finds a new identity, there aren't many people who get excited about it any more, except the discoverer.
However, I barely know any identities about Pascal's triangle. The only two that I know are that the second diagonal contains all the counting numbers and that $C(n,k)$ is the entry in Pascal's triangle that is down in the $n^{th}$ row $k$ spaces along.
What are some of the identities and theorems that are about Pascal's triangle?
Note: I found this question, but it only has one answer, so I asking this now, because I want to know more.
Well, among the identities/theorems, the one that I have always found the coolest is that the sum of the $n$-th shallow diagonal is the $n$-th Fibonacci number (see below).
Some other cool relations follow directly from combinatorial principles - like fact that the sum of the elements in the $n$-th row is $2^n$, and the triangle is vertically symmetric (as ${n\choose r}={n\choose n-r}$).
Yet another $\textit{awesome}$ connection comes from examining what happens when only considering triangle modulo $2$. As you zoom out on the triangle, the regions of odd and even begin to look more and more like Sierpinski's triangle.
I've included the classic graphic of the Fibonacci numbers in Pascal's triangle, and a picture of Sierpinski's triangle.
EDIT: My images don't seem to be working. About halfway down this page is the classic Fibonacci-Pascal diagram, and here is the link to the Sierpinski triangle wikipedia page which has a multitude of cool pictures.