# Pascal's pyramid

The first five levels of Pascal's pyramid

The Pascal pyramid is the three-dimensional generalization of the Pascal triangle . It contains the third-order multinomial coefficients (trinomial coefficient), i.e. H. the coefficients of are at level n +1. As in the Pascal triangle, the Pascal pyramid begins with a single 1 on the top level (the "top" of the pyramid ). Each additional number is the sum of the three numbers above it. All the special properties of the Pascal triangle (see e.g. Sierpinski triangle , symmetry ) can also be applied to the Pascal pyramid. ${\ displaystyle (a + b + c) ^ {n}}$

## Alternative construction

The trinomial coefficients are given by

${\ displaystyle {\ frac {(i + j + k)!} {i! \, j! \, k! \;}}}$ With ${\ displaystyle \; i + j + k = n \ ,.}$

The identity

${\ displaystyle {\ frac {(i + j + k)!} {i! \, j! \, k!}} = {\ frac {(i + j + k)!} {(i + j)! \, k!}} \ cdot {\ frac {(i + j)!} {i! \, j!}}}$

suggests the following construction rule for the ( n + 1) th level:

1. First, form the three sides of the triangle. These correspond to the ( n + 1) -th line in Pascal's triangle.
2. Now fill the m th line with the entries from the m th line of Pascal's triangle, multiplied by the factor already entered on the sides.

## The first seven levels

1st level

                     1


2nd level

                     1
1     1


3rd level

                     1
2     2
1     2     1


4th level

                     1
3      3
3     6      3
1     3      3     1


5th level

                     1
4      4
6    12      6
4    12     12     4
1     4     6      4     1


6th level

                      1
5     5
10    20    10
10    30    30    10
5    20    30    20     5
1     5    10    10     5     1


7th level

                      1
6     6
15    30    15
20    60    60    20
15    60    90    60    15
6    30    60    60    30     6
1     6    15    20    15     6     1


## properties

• The sum of all numbers on level n is:${\ displaystyle 3 ^ {n-1}}$
• The sum of all numbers from the first to the nth level is:${\ displaystyle {\ frac {3 ^ {n} -1} {2}}}$

### Connection with the Sierpinski tetrahedron

If the Pascal tetrahedron distinguishes between even and odd numbers, there is a connection with the Sierpinski tetrahedron . The even numbers correspond to the gaps in the Sierpinski tetrahedron. Here, have levels are taken into account in order to obtain th iteration in the construction of the Sierpinski tetrahedron. ${\ displaystyle 2 ^ {a}}$${\ displaystyle a}$

## generalization

Similarly, the -dimensional Pascal Simplex can be defined from the other multinomial coefficients. ${\ displaystyle n}$