The number in each row is the sum of the two numbers just above it. Increase the number of rows, and color the odd and even numbers to see how the pattern appears.
These numbers are actually a count of outcomes. The number of ways to get exactly k heads when you flip a coin n times is exactly the k-th number in row n of Pascal's triangle. Flip a coin 4 times, and the number of ways to get 0, 1, 2, 3, or 4 heads is 1, 4, 6, 4, 1 respectively — which is exactly the fourth row of Pascal's triangle.
Click the "Color odd/even" tab and something remarkable appears. Just by coloring the odd numbers, a triangular pattern that keeps repeating itself (Sierpiński's triangle) shows up. Increase the number of rows to the maximum and watch this pattern get more and more distinct.
It's no coincidence that the sum of each row always comes out to a power of 2, either. The rule that builds each row by adding the two numbers just above it exactly doubles the total sum every single row.