Pascal's triangle

Pascal's triangle is an infinite triangular array of the binomial coefficients, named after French mathematician Blaise Pascal.

The rows of Pascal's triangle are conventionally enumerated starting with row $n = 0$ at the top. The entries in each row are numbered from the left beginning with $k = 0$ .

Each entry can be expressed in the form $(\binom{n}{k})$ where $n$ is the row number and $k$ is the column number.

These values are the binomial coefficients, i.e. we can compute each entry using the formula

(\binom{n}{k}) = \frac{n!}{k! (n - k)!}

The triangle may be recursively constructed in the following manner:

In row 0, there is a unique nonzero entry 1.
Each entry of each subsequent row is constructed by adding the number above and to the left with the number above and to the right, treating blank entries as 0.

This is expressed via Pascal's rule. Formally,

(\binom{n}{k}) = (\binom{n - 1}{k - 1}) + (\binom{n - 1}{k})