Tables

Arithmetical TriangleThis is often called the Pascal Triangle, since Pascal demonstrated many of its intriguing properties, but it was known by earlier mathematicians; the oldest known version is actually Chinese. Each line consists of a 1 at the edges (so to speak), with any other value derived by adding together the two numbers above it in the triangle. The result, for row n, is the binomial coefficients for the expansion of (x + y)*n. These are of fundamental importance in the calculation of binomial or Bernoullian probabilities, and in combinatorics generally.

Each row may conveniently be named for the second digit from the left in that row. Since there is no second digit in the top row, n = 0 in that case. These row names are given in the lefthand column of the diagram.

Triangle

0 1 1 1 1 2 1 2 1 3 1 3 3 1 4 1 4 6 4 1 5 1 5 10 10 5 1 6 1 6 15 20 15 6 1 7 1 7 21 35 35 21 7 1 8 1 8 28 56 70 56 28 8 1 9 1 9 36 84 126 126 84 36 9 1 10 1 10 45 120 210 252 210 120 45 10 1 11 1 11 55 165 330 462 462 330 165 55 11 1 12 1 12 66 220 495 792 924 792 495 220 66 12 1

The sum of all numbers in any row n equals 2*n. This fact leads to some helpfully short formulas for authorship and analogous problems in stylistic analysis.

For a less pretty version, see the Right Arithmetical Triangle.

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