As said before, the nth row is the binomial coefficients of (x+y)^n.

Pascal's triangle is 0-index based, that is the first row is the 0th row. The first term in the any row, is the 0th.

Next we find that:

(m, n) = m choose n.

(m, n) = (m-1, n-1) + (m-1, n) That is, that the sum of two adjacent terms is equal to the term directly below and inbetween.

Another interesting occurence is that the sum of the terms in the nth row is equal to 2^n.

Here are the first 5 rows:

........1

......1..1.

....1..2...1

..1..3..3...1

1..4..6..4..1

Probably the most obvious observation is the fact that each row is symmetric.

On the first diagonal (in either direction), we get 1's.

On the 2nd diagonal, we get the counting numbers: 1, 2, 3, etc.

On the 3rd diagonal, we get the triangular numbers: 1, 3, 6, etc.

On the 4th diagonal, we get the sum of triangular numbers: 1, 4, 10, etc.

Play around with it, and you will find some pretty interesting things.