Pascal's triangle

Pascal's triangle is a triangular stack of numbers where every entry is the sum of the two numbers directly above it, with 1's running down both edges. Start with a single 1 at the top, and the whole triangle grows from that one addition rule.

Its rows (counting the top as row 0) are exactly the binomial coefficients: row n holds C(n, 0), C(n, 1), ..., C(n, n), which are the coefficients in the expansion of (a + b)^n. So you can read off (a + b)^4 = 1, 4, 6, 4, 1 straight from row 4.

The triangle hides many patterns: each row is symmetric, the row sums double each time (they equal 2^n), and the shallow diagonals add up to the Fibonacci numbers. In Chinese mathematics the same array appears centuries earlier as Yang Hui's triangle.