Pascal's Pyramid - Addition of Coefficients Between Layers

Addition of Coefficients Between Layers

The numbers on every layer (n) of the Tetrahedron are the sum of the three adjacent numbers in the layer (n−1) "above" it. This relationship is rather difficult to see without intermingling the layers. Below are italic Layer 3 numbers interleaved among bold Layer 4 numbers:

The relationship is illustrated by the lower, central number 12 of the 4th Layer. It is "surrounded" by three numbers of the 3rd Layer: 6 to the "north", 3 to the "southwest", 3 to the "southeast". (The numbers along the edge have only two adjacent numbers in the layer "above" and the three corner numbers have only one adjacent number in the layer above, which is why they are always "1". The missing numbers can be assumed as "0", so there is no loss of generality.) This relationship between adjacent layers is not a magical coincidence. Rather, it comes about through the two-step trinomial expansion process.

Continuing with this example, in Step 1, each term of (A + B + C)3 is multiplied by each term of (A + B + C)1. Only three of these multiplications are of interest in this example:

(The multiplication of like variables causes the addition of exponents; e.g.: D1 × D2 = D3.)

Then, in Step 2, the summation of like terms (same variables and exponents) results in: 12A1B2C1, which is the term of (A + B + C)4; while 12 is the coefficient of the 4th Layer of the Tetrahedron.

Symbolically, the additive relation can be expressed as:

where C(x,y,z) is the coefficient of the term with exponents x, y, z and x+y+z = n is the layer of the Tetrahedron.

This relationship will work only if the trinomial expansion is laid out in the non-linear fashion as it is portrayed in the section on the "trinomial expansion connection".