1911 Encyclopædia Britannica/Figurate Numbers

FIGURATE NUMBERS, in mathematics. If we take the sum of n terms of the series 1 + 1 + 1 + . . ., i.e. n, as the nth term of a new series, we obtain the series 1 + 2 + 3 + . . ., the sum of n terms of which is 1/2n · n + 1. Taking this sum as the nth term, we obtain the series 1 + 3 + 6 + 10 + . . ., which has for the sum of n terms n (n + 1) (n + 2) / 3!^[1] This sum is taken as the nth term of the next series, and proceeding in this way we obtain series having the following nth terms:—

1, n, n(n + 1)/2!, n(n + 1)(n + 2)/3!, ...n(n+1) ...(n + r − 2)/(r − 1)!.

The numbers obtained by giving n any value in these expressions are of the first, second, third, . . . or r th order of figurate numbers.

Pascal treated these numbers in his Traité du triangle arithmetique (1665), using them to develop a theory of combinations and to solve problems in probability. His table is here shown in its simplest form. It is to be noticed that each number is the sum of the numbers immediately above and to the left of it; and that the numbers along a line, termed a base, which cuts off an equal number of units along the top row and column are the coefficients in the binomial expansion of (1 + x)^r−1, where r represents the number of units cut off.

1. The notation n! denotes the product 1 · 2 · 3 · . . . n, and is termed “factorial n.”