Tables
Arithmetical Triangle

This 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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