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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]
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
"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
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 ratio of the factorial!, that counts all permutations of an ordered set S with cardinality, and the subfactorial (a.k.a. the derangement function) !, which counts the amount of permutations where no element appears in its original position, tends to as grows.
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.
De Polignac's formula; Difference operator; Difference polynomials; Digamma function; Egorychev method; Erdős–Ko–Rado theorem; Euler–Mascheroni constant; Faà di Bruno's formula; Factorial; Factorial moment; Factorial number system; Factorial prime; Gamma distribution; Gamma function; Gaussian binomial coefficient; Gould's sequence ...
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 ...