Pick the number that you want to start your triangular number diagram at. Usually this is the number 1 for simplicity, but you can pick any number you want. Write this number at the top of your paper or on whatever you're writing your diagram.
Copy the first number you wrote down below it, twice. These three numbers (all the same) should form a triangle. The rest of the diagram will continue down along this shape. Copy the numbers along the second row into the third row on the right and left side and then add these two numbers together for the middle number. This should form a triangle with rows of one number, two numbers and three numbers respectively.
Continue to have your same starter number expand the outer edges of the triangle on the right and left side. Each number in a row should be comprised of the two numbers to the upper left and right of it. If you start at "1" the rows will follow a specific pattern. The first row will be "1," then second will be "1,1," the third will be "1, 2, 1," the fourth will be "1, 3, 3, 1" and so on.
The triangular number diagram is known more commonly as Pascal's triangle and is most useful in determining binomial expansion coefficients. These are the numbers you get if you have two numbers in a parenthesis raised to a certain power. The second row corresponds to the first power, the third row is for if you square the term, the fourth corresponds to cubing, and so on.