Pascal's pyramid

The Pascal's pyramid is the three dimensional generalization of Pascal's triangle. It contains the multinomial third order ( Trinomialkoeffizient ), ie the coefficients of are at level n 1. As in Pascal's triangle is Pascal's pyramid with a single 1 at the top level ( the "top" of the pyramid ). Any other number is the sum of the three standing on their numbers. All the special properties of Pascal's triangle ( see, eg, Sierpinski triangle, symmetry ) can be applied in context on Pascal's pyramid.

Alternative construction

The Trinomialkoeffizienten are given by

The identity

Suggests the following design specification for the (n 1 )-th level:

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 level 6

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 of the plane is n:
• The sum of all the numbers of the first to the n-th level is:

Related to the Sierpinski tetrahedron

If just in Pascal's tetrahedron and distinguished odd numbers, there is a connection with the Sierpinski tetrahedron. The even numbers correspond to the gaps in the Sierpinski tetrahedron. In this case, levels must be taken into account in order to obtain the - th iteration step in the construction of the Sierpinski tetrahedron.

Generalization

Analogously, define the further from the multinomial -dimensional Pascal's simplex.