Tables
Right Arithmetical Triangle

The standard version of this triangle shows more clearly how each number is derived, but this format is slightly handier for displaying certain properties. In the lefthand column, as before, is given the row name, and in the righthand column, the corresponding sum of all numbers in that row. For any row n, this sum turns out to be equal to 2*n. That is, a power series (such as 2*n) can be derived not only by multiplication, but also by a complicated process of addition of strings of related numbers.

Triangle

n
Binomial Coefficients for (p + q)*n
2*n
0
1
1
1
1
1
2
2
1
2
1
4
3
1
3
3
1
8
4
1
4
6
4
1
16
5
1
5
10
10
5
1
32
6
1
6
15
20
15
6
1
64
7
1
7
21
35
35
21
7
1
128
8
1
8
28
56
70
56
28
8
1
256
9
1
9
36
84
126
126
84
36
9
1
512
10
1
10
45
120
210
252
210
120
45
10
1
1024
11
1
11
55
165
330
462
462
330
165
55
11
1
2048
12
1
12
66
220
495
792
924
792
495
220
66
12
1
4096
13
1
13
78
286
715
1287
1716
1716
1287
715
286
78
13
1
8192
14
1
14
91
364
1001
2002
3003
3432
3003
2002
1001
364
91
14
1
16384
15
1
15
105
455
1365
3003
5005
6435
6435
5005
3003
1365
455
105
15
1
32768
16
1
16
120
560
1820
4368
8008
11440
12870
11440
8008
4368
1820
560
120
16
1
65536
17
1
17
136
680
2380
6188
12376
19448
24310
24310
19448
12376
6188
2380
680
136
17
1
131072
18
1
18
153
816
3060
8568
18564
31824
43758
48620
43758
31824
18564
8568
3060
816
153
18
1
262144

Postscript

The diagonals on the basic triangle here appear as orthogonals, and it is thus easier to point out some of the curious features of the former diagonals, now the columns on the present chart.

  • Column 1 (1, 1, 1, . . ) is simply the elementary number 1, repeated indefinitely.
  • Column 2 (1, 2, 3, . . ) may thought of as the cumulative sum of Column 1. It is the familiar series of natural numbers.
  • Column 3 (1, 3, 6, 10, . . ) is in turn the cumulative sum of Column 2. This is what we have elsewhere called a summarial: the sum of the natural numbers up to a particular point. It is the additive analogue of the factorial (the product of the natural numbers, up to a particular point). These are also the triangular numbers.
  • Column 4 (1, 4, 10, 20, . . ) is the tetrahedral numbers.
  • Column 5 (1, 5, 15, 35, . . ) is another series of figurate numbers. Later columns are successively higher order figurate numbers.

The "strings of related numbers" in question are thus all possible series of figurate numbers, beginning with the minimal 1, 1, 1 (a figurate point), each offset one position from the last, and summed in columns (here, rows).

It was perhaps not to be expected from this definition that the sums in question would yield successive powers of 2.

For other remarkable properties of the triangle, viewers should directly consult Pascal, who is the master in this area.

Back to Tables Page

Contact The Project / Exit to Resources Page