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.

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