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Factorials grow faster than exponential functions, but much more slowly than double exponential functions. However, tetration and the Ackermann function grow faster. See Big O notation for a comparison of the rate of growth of various functions. The inverse of the double exponential function is the double logarithm log(log(x)).
Just as triangular numbers sum the numbers from to , and factorials take their product, the exponential factorial exponentiates. The exponential factorial is defined recursively as =, =. For example, the exponential factorial of 4 is = These numbers grow much more quickly than regular factorials. [95] Falling factorial
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 .
An alternative notation for the rising factorial () is the less common () +. When () + is used to denote the rising factorial, the notation () is typically used for the ordinary falling factorial, to avoid confusion. [3]
The exponential factorials grow much more quickly than regular factorials or even hyperfactorials. The number of digits in the exponential factorial of 6 is approximately 5 × 10 183 230. The sum of the reciprocals of the exponential factorials from 1 onwards is the following transcendental number:
Growth rates may also be faster than exponential. In the most extreme case, when growth increases without bound in finite time, it is called hyperbolic growth . In between exponential and hyperbolic growth lie more classes of growth behavior, like the hyperoperations beginning at tetration , and A ( n , n ) {\displaystyle A(n,n)} , the diagonal ...
These numbers grow more quickly than polynomials but more slowly than exponentials. As well as in the symmetries of trees, they arise as the numbers of transitive orientations of comparability graphs [ 3 ] and in the problem of finding factorials that can be represented as products of smaller factorials.
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 ) ! !