Second Column and Diagonal of Pascal's Triangle consist of Triangular Numbers

Theorem
The $2$nd column and $2$nd diagonal of Pascal's triangle consists of the set of triangular numbers.

Proof
Recall Pascal's triangle:

By definition, the entry in row $n$ and column $m$ contains the binomial coefficient $\dbinom n m$.

Thus the $2$nd column contains all the elements of the form $\dbinom n 2$.

The $m$th diagonal consists of the elements in column $n - m$.

Thus the $m$th diagonal contains the binomial coefficients $\dbinom n {n - m}$.

By Symmetry Rule for Binomial Coefficients:
 * $\dbinom n {n - m} = \dbinom n m$

Thus the $2$nd diagonal also contains the binomial coefficients $\dbinom n 2$.

By Binomial Coefficient with Two: Corollary, the triangular numbers are precisely those numbers of the form $\dbinom n 2$.

Hence the result.