Above are coefficients of (a+b)^n for n= 0 to 5.
The two dimension simplex is the triangle. The numbers in the d=2 column
are triangle numbers.
Triangle numbers can be arranged in this way:
They are sums of consecutive integers starting from 1.
Tetrahedron numbers are sums of consecutive triangle numbers starting from 1.
I call the set of positive integers line numbers because they can be
arranged in a line.
The number sixteen, for example could be arranged thus: 0000000000000000
I think of one as the point number.
I believe the dth column of Pascal's triangle shows consecutive d-simplex
This assumes a point is a 0 dimension simplex and a line segment is a 1 dimension simplex.
Math Forum's Pascal's Triangle Page
Sierpinksi Pattern's in Pascal's Triangle