Columns of Pascal's Triangle contain Simplicial Polytopic Numbers

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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.


Proof


Sources