Columns of Pascal's Triangle contain Simplicial Polytopic Numbers

Theorem
The columns of Pascal's triangle contain the simplicial polytopic numbers:


 * Column $0$: repeated instances of number $1$


 * Column $1$: the (strictly) positive integers


 * Column $2$: the triangular numbers


 * Column $3$: the tetrahedral numbers


 * Column $4$: the pentatope numbers

and so on.