Tables

Right Arithmetical TriangleThe 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

nBinomial 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 262144Postscript

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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