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The word "factorial" (originally French: factorielle) was first used in 1800 by Louis François Antoine Arbogast, [18] in the first work on Faà di Bruno's formula, [19] but referring to a more general concept of products of arithmetic progressions. The "factors" that this name refers to are the terms of the product formula for the factorial. [20]
These are counted by the double factorial 15 = (6 − 1)‼. In mathematics, the double factorial of a number n, denoted by n‼, is the product of all the positive integers up to n that have the same parity (odd or even) as n. [1] That is,
The falling factorial can be extended to real values of using the gamma function provided and + are real numbers that are not negative integers: = (+) (+) , and so can the rising factorial: = (+) . Calculus
A more efficient method to compute individual binomial coefficients is given by the formula = _! = () (()) () = = +, where the numerator of the first fraction, _, is a falling factorial. This formula is easiest to understand for the combinatorial interpretation of binomial coefficients.
"Stirling_formula", Encyclopedia of Mathematics, EMS Press, 2001 [1994] Peter Luschny, Approximation formulas for the factorial function n! Weisstein, Eric W., "Stirling's Approximation", MathWorld; Stirling's approximation at PlanetMath
This may be expressed as stating that, in the formula for () as a product of factorials, omitting one of the factorials (the middle one, ()!) results in a square product. [4] Additionally, if any n + 1 {\displaystyle n+1} integers are given, the product of their pairwise differences is always a multiple of s f ( n ) {\displaystyle {\mathit {sf ...
1. Factorial: if n is a positive integer, n! is the product of the first n positive integers, and is read as "n factorial". 2. Double factorial: if n is a positive integer, n!! is the product of all positive integers up to n with the same parity as n, and is read as "the double factorial of n". 3.
In mathematics, especially in combinatorics, Stirling numbers of the first kind arise in the study of permutations. In particular, the unsigned Stirling numbers of the first kind count permutations according to their number of cycles (counting fixed points as cycles of length one).