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Much of the mathematics of the factorial function was developed beginning in the late 18th and early 19th centuries. Stirling's approximation provides an accurate approximation to the factorial of large numbers, showing that it grows more quickly than exponential growth.
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.
The factorial number system is sometimes defined with the 0! place omitted because it is always zero (sequence A007623 in the OEIS). In this article, a factorial number representation will be flagged by a subscript "!". In addition, some examples will have digits delimited by a colon. For example, 3:4:1:0:1:0! stands for
Fuss–Catalan number; Central binomial coefficient; Combination; Combinatorial number system; 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 ...
This fact can also be applied to finite series of rational functions, allowing the result to be computed in constant time even when the series contains a large number of terms. Exponential function [ edit ]
In mathematics, and more specifically number theory, the hyperfactorial of a positive integer is the product of the numbers of the form from to . Definition [ edit ]
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
In mathematics, the (signed and unsigned) Lah numbers are coefficients expressing rising factorials in terms of falling factorials and vice versa. They were discovered by Ivo Lah in 1954. [ 1 ] [ 2 ] Explicitly, the unsigned Lah numbers L ( n , k ) {\displaystyle L(n,k)} are given by the formula involving the binomial coefficient