Collection of patterns in Pascal's triangle
Every number is the sum of the two numbers above it (source).
The second diagonal contains the counting numbers (source).
The third diagonal contains the triangular numbers (source).
It has a vertical line of symmetry (source).
Each row sums to a power of 2 (source).
All numbers in the row, excluding the two exterior 1s, are divisible by the second number in the row, if and only if the second number is prime. For the diagonals we have this pattern:
If all the even numbers are circled, Sierpinski's triangle appears (source).
The sum of certain diagonals yield Fibonacci numbers:
The hockey-stick identity: