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This is a list of factorial and binomial topics in mathematics. See also binomial (disambiguation). Abel's binomial theorem; Alternating factorial; Antichain; Beta function; Bhargava factorial; Binomial coefficient. Pascal's triangle; Binomial distribution; Binomial proportion confidence interval; Binomial-QMF (Daubechies wavelet filters ...
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 symbols are collectively called factorial powers. [2] The Pochhammer symbol, introduced by Leo August Pochhammer, is the notation (), where n is a non-negative integer. It may represent either the rising or the falling
For =, the sum of the factorials of the digits is simply the number of digits in the base 2 representation since ! =! =. A natural number n {\displaystyle n} is a sociable factorion if it is a periodic point for SFD b {\displaystyle \operatorname {SFD} _{b}} , where SFD b k ( n ) = n {\displaystyle \operatorname {SFD} _{b}^{k}(n)=n} for a ...
Factorial experiments are described by two things: the number of factors, and the number of levels of each factor. For example, a 2×3 factorial experiment has two factors, the first at 2 levels and the second at 3 levels. Such an experiment has 2×3=6 treatment combinations or cells.
Comparison of Stirling's approximation with the factorial In mathematics , Stirling's approximation (or Stirling's formula ) is an asymptotic approximation for factorials . It is a good approximation, leading to accurate results even for small values of n {\displaystyle n} .
Just as the factorials can be continuously interpolated by the gamma function, the superfactorials can be continuously interpolated by the Barnes G-function. [ 2 ] According to an analogue of Wilson's theorem on the behavior of factorials modulo prime numbers, when p {\displaystyle p} is an odd prime number s f ( p − 1 ) ≡ ( p − 1 ) ! !
Multiplicative partitions of factorials are expressions of values of the factorial function as products of powers of prime numbers. They have been studied by Paul Erdős and others. [1] [2] [3] The factorial of a positive integer is a product of decreasing integer factors, which can in turn be factored into prime numbers.
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