Pascal’s triangle is filled with a wealth of interesting patterns, something I never learnt during secondary school when first introduced to this mathematical object. In this post, inspired by a Numberphile video, I want to show you some of these patterns that you might have never seen before.

Firstly, **what is Pascal’s Triangle?**

Pascal’s triangle is a triangular array of the binomial coefficients.

To construct Pascal’s triangle, start with the two top rows, which are **1** and **1 1**. To find any number in the next row, add the two numbers above it.

At the beginning and end of each row, where there is only one number above, write 1.

### Symmetric

Pascal’s triangle is symmetric, a fact which follows directly from the formula for the binomial coefficient:

It can also be seen by how the triangle is constructed, i.e. “*by the interpretation of the entries as the number of ways to get from the top to a given spot in the triangle*“.

### Sum of entries in row *n* equals *2*^{n}

This fact can be proved by induction. The main point of the argument is that each entry in row *n* is added to two entries below. This gives us:

Hence, the sum of entries in row n+1 is twice the sum of entries in row n.

### Hockey Stick Pattern

In Pascal’s words: “*In every arithmetical triangle each cell is equal to the sum of all the cells of the preceding row from its column to the first, inclusive”. *This is easier to explain using a diagram:

Mathematically, this can be proved by induction and is denoted in the following way:

### Fibonacci Numbers

To see this, let us first rearrange Pascal’s triangle.

The successive Fibonacci numbers are the sums of the entries on the marked diagonals:

1 = 1

1 = 1

2 = 1 + 1

3 = 1 + 2

5 = 1 + 3 + 1

8 = 1 + 4 + 3

13 = 1 + 5 + 6 + 1

etc.

### e

Recently, the Harlan brothers highlighted that *e* is hidden in Pascal’s triangle. This was discovered by considering *products *rather than *sums*.

Denoting Sn as the product of the terms in the *n*th row, as *n* tends to infinity, we find that:

Click here for the derivation.

### Catalan Numbers

Catalan numbers can be found in Pascal’s triangle in a few ways, for example:

- If you take each ‘middle’ element and subtract its adjacent entry, you get a Catalan number.

- If you take a middle element and divide it by its position in the list of middle terms (e.g. divide the 5th middle term by 5), you will get a Catalan number.

There are many more patterns, and I encourage you to find out more about them! M x