Pascal's pyramid

In mathematics, Pascal's Pyramid is a three-dimensional arrangement of the trinomial numbers, which are the coefficients of the trinomial expansion and the trinomial distribution. Pascal's Pyramid is the three-dimensional analog of the two-dimensional Pascal's Triangle, which contains the binomial numbers and relates to the binomial expansion and the binomial distribution. The binomial and trinomial numbers, coefficients, expansions, and distributions are subsets of the multinomial constructs with the same names. Pascal's Pyramid is more precisely called "Pascal's Tetrahedron", since it has four triangular surfaces. (The pyramids of ancient Egypt had five surfaces: a square base and four triangular sides.)

Construction

Because the pyramid is a three-dimensional object it is difficult to display it on a piece of paper or a computer screen. So the pyramid is divided into a number of slices, or levels, or layers, or floors. The top layer (the apex) is labeled Layer 0. Other layers can be thought of as overhead views of the pyramid with the previous layers removed. The first six layers are as follows:

Layer 0

Layer 1

                       1       1
                           1

Layer 2

                   1       2       1
                       2       2
                           1

Layer 3

               1       3       3       1
                   3       6       3
                       3       3
                           1

Layer 4

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

Layer 5

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

Sum of numbers in layer above

Here is a diagram showing how level five arises from level four. The black dots stand for 12 (level four) and the red dots for sites on level five which are the sum of the adjacent numbers from level four.

Trinomial construction

Arbitrary layer n can be constructed by enumerating all corresponding trinomial coefficients (or a sixth of them thanks to the symmetry).

Example for n = 3:

\textstyle {3 \choose 0,3,0}

\textstyle {3 \choose 1,2,0}\ {3 \choose 0,2,1}

\textstyle {3 \choose 2,1,0}\ {3 \choose 1,1,1}\ {3 \choose 0,1,2}

\textstyle {3 \choose 3,0,0}\ {3 \choose 2,0,1}\ {3 \choose 1,0,2}\ {3 \choose 0,0,3}

Exponentiational construction

Arbitrary layer n can be obtained in a single step using the following formula:

\left(b^{d\left(n+1\right)}+b^{d}+1\right)^{n},

where b is the radix and d is the number of digits of any of the central multinomial coefficients, that is

\textstyle d=1+\left\lfloor \log _{b}{n \choose k_{1},k_{2},k_{3}}\right\rfloor ,\ \sum _{i=1}^{3}{k_{i}}=n,\ \left\lfloor {\frac {n}{3}}\right\rfloor \leq k_{i}\leq \left\lceil {\frac {n}{3}}\right\rceil ,

then wrapping the digits of its result by d(n+1), spacing by d and removing leading zeros.

This method generalised to arbitrary dimension can be used to obtain slices of any Pascal's simplex.

Examples

For radix b = 10, n = 5, d = 2:

\textstyle \left(10^{12}+10^{2}+1\right)^{5}

= 1000000000101⁵
= 1000000000505000000102010000010303010000520302005010510100501

              1                     1                     1
   000000000505     00 00 00 00 05 05     .. .. .. .. .5 .5
   000000102010     00 00 00 10 20 10     .. .. .. 10 20 10
~  000010303010  ~  00 00 10 30 30 10  ~  .. .. 10 30 30 10
   000520302005     00 05 20 30 20 05     .. .5 20 30 20 .5
   010510100501     01 05 10 10 05 01     .1 .5 10 10 .5 .1

 wrapped by d(n+1)     spaced by d      leading zeros removed

For radix b = 10, n = 20, d = 9:

\textstyle \left(10^{189}+10^{9}+1\right)^{20}

Pascal's pyramid layer #20.

Features

There is three-way symmetry of the numbers in each layer.
The number of terms in the n^th layer is the n^th triangular number: (n + 1) × (n + 2) / 2.
The sum of the values of the numbers in the n^th layer is 3ⁿ.
Each number in any layer is the sum of the three adjacent numbers in the layer above.
Each number in any layer is a simple whole number ratio of the adjacent numbers in the same layer.
Each number in any layer is a coefficient of the trinomial distribution and the trinomial expansion. This non-linear arrangement makes it easier to:
- display the trinomial distribution in a coherent way;
- calculate the coefficients of the trinomial distribution;
- calculate the numbers of any tetrahedron layer.
The numbers along the three edges of the n^th layer are the numbers of the n^th line of Pascal's Triangle. And almost all the features listed above have parallels with Pascal's Triangle and Multinomial Coefficients.

Sum of all coefficients of a layer

From the formula for a sum of multinomial coefficients of a layer n follows that for the Pascal's pyramid, where the m = 3, this sum equals

\sum _{k_{1},k_{2},k_{3}}{n \choose k_{1},k_{2},k_{3}}=3^{n}.

For example:

 1            1                    1                                1
---         1   1                2   2                            3   3
 1 = 3⁰    -------             1   2   1                        3   6   3
            1 + 1 + 1 = 3¹    -----------                     1   3   3   1
                               1 + 2 + 1 + 2 + 2 + 1 = 3²    ---------------
                                                              1 + 3 + 3 + 1 + 3 + 6 + 3 + 3 + 3 + 1 = 3³

Sum of coefficients of a layer by rows

Summing the numbers in each row of a layer n of Pascal's pyramid gives

\left(b^{d}+2\right)^{n},

where b is the radix and d is the number of digits of the sum of the 'central' row (the one with the greatest sum).

For radix b = 10:

 1 ~ 1    \ 1  ~ 1      \ 1   ~ 1          \ 1    ~  1               \ 1     ~  1
---      1 \ 1 ~ 2   \ 2 \ 2  ~ 4       \ 3 \ 3   ~ 06            \ 4 \ 4    ~ 08
 1       -----      1 \ 2 \ 1 ~ 4    \ 3 \ 6 \ 3  ~ 12         \ 6 \12 \ 6   ~ 24
         1   2      ---------       1 \ 3 \ 3 \ 1 ~ 08      \ 4 \12 \12 \ 4  ~ 32
                    1   4   4       -------------          1 \ 4 \ 6 \ 4 \ 1 ~ 16
                                    1  06  12  08         ------------------
                                                           1  08  24  32  16

12⁰       12¹          12²               102³                     102⁴

Sum of coefficients of a layer by columns

Summing the numbers in each column of a layer n of Pascal's pyramid gives

\left(b^{2d}+b^{d}+1\right)^{n},

where b is the radix and d is the number of digits of the sum of the 'central' column (the one with the greatest sum).