Pascal's triangle

In Pascal's triangle, each number is the sum of the two numbers directly above it.

Pascal's triangle determines the coefficients which arise in binomial expansions. For example, consider the expansion

Pascal's triangle has many properties and contains many patterns of numbers.

Each frame represents a row in Pascal's triangle. Each column of pixels is a number in binary with the least significant bit at the bottom. Light pixels represent ones and the dark pixels are zeroes.

The diagonals of Pascal's triangle contain the figurate numbers of simplices:

There are simple algorithms to compute all the elements in a row or diagonal without computing other elements or factorials.

A level-4 approximation to a Sierpinski triangle obtained by shading the first 32 rows of a Pascal triangle white if the binomial coefficient is even and black if it is odd.
Pascal's triangle overlaid on a grid gives the number of distinct paths to each square, assuming only rightward and downward movements are considered.

Due to its simple construction by factorials, a very basic representation of Pascal's triangle in terms of the matrix exponential can be given: Pascal's triangle is the exponential of the matrix which has the sequence 1, 2, 3, 4, ... on its subdiagonal and zero everywhere else.

each of which is built upon elements of one fewer dimension from the original triangle

That is, choose a pair of numbers according to the rules of Pascal's triangle, but double the one on the left before adding. This results in:

compose the 4th row of the triangle, with alternating signs. This is a generalization of the following basic result (often used in electrical engineering):

Another option for extending Pascal's triangle to negative rows comes from extending the other line of 1s:

Also, just as summing along the lower-left to upper-right diagonals of the Pascal matrix yields the Fibonacci numbers, this second type of extension still sums to the Fibonacci numbers for negative index.