Coefficients of (a+b)^n

d=0 d=1 d=2 d=3 d=4 d=5
n=0 1
n=1 1 1
n=2 1 2 1
n=3 1 3 3 1
n=4 1 4 6 4 1
n=5 1 5 10 10 5 1

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 numbers.
This assumes a point is a 0 dimension simplex and a line segment is a 1 dimension simplex.

Swarthmore Math Forum's Pascal's Triangle Page
Sierpinksi Pattern's in Pascal's Triangle