Pascal's triangle

Description

Pascal's triangle is the triangular array of the binomial coefficients. Its rows are numbered $n = 0, 1, 2, \dots$ from the top. Row $n$ has $n + 1$ entries, $\binom{n}{0}, \binom{n}{1}, \dots, \binom{n}{n}$, from left to right.

The first five rows are these. $$\begin{array}{ccccccccc} & & & & 1 & & & & \\ & & & 1 & & 1 & & & \\ & & 1 & & 2 & & 1 & & \\ & 1 & & 3 & & 3 & & 1 & \\ 1 & & 4 & & 6 & & 4 & & 1 \end{array}$$

Each row starts and ends with $1$. Each other entry is the sum of the two entries next to it in the row above, by Pascal's rule.

Remarks

Row $n$ lists the coefficients of $(x + y)^n$, by the binomial theorem. The sum of row $n$ is $2^n$. This is the sum of binomial coefficients. Each row is symmetric, by the symmetry of binomial coefficients.

Explanations

L1 human

Start with a $1$ at the top. Each new row starts and ends with $1$, and each inside number is the sum of the two numbers above it. Row 4 is $1, 4, 6, 4, 1$, and these are the coefficients of $(x + y)^4$.