Exponential factorial
Encyclopedia
An exponential factorial is a positive integer n raised to the power
of n − 1, which in turn is raised to the power of n − 2, and so on and so forth, that is,
The exponential factorial can also be defined with the recurrence relation
The first few exponential factorials are 1, 1, 2, 9, 262144, etc. . So, for example, 262144 is an exponential factorial since
The exponential factorials grow much more quickly than regular factorial
s or even hyperfactorials. The exponential factorial of 5 is 5262144 which is approximately 6.206069878660874 × 10183230.
The sum of the reciprocals of the exponential factorials from 1 onwards is the irrational number
1.6111149258083767361111... .
Like tetration
, there is currently no accepted method of extension of the exponential factorial function to real
and complex
values of its argument, unlike the factorial
function, for which such an extension is provided by the gamma function
.
Exponentiation
Exponentiation is a mathematical operation, written as an, involving two numbers, the base a and the exponent n...
of n − 1, which in turn is raised to the power of n − 2, and so on and so forth, that is,
The exponential factorial can also be defined with the recurrence relation
Recurrence relation
In mathematics, a recurrence relation is an equation that recursively defines a sequence, once one or more initial terms are given: each further term of the sequence is defined as a function of the preceding terms....
The first few exponential factorials are 1, 1, 2, 9, 262144, etc. . So, for example, 262144 is an exponential factorial since
The exponential factorials grow much more quickly than regular factorial
Factorial
In mathematics, the factorial of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n...
s or even hyperfactorials. The exponential factorial of 5 is 5262144 which is approximately 6.206069878660874 × 10183230.
The sum of the reciprocals of the exponential factorials from 1 onwards is the irrational number
Irrational number
In mathematics, an irrational number is any real number that cannot be expressed as a ratio a/b, where a and b are integers, with b non-zero, and is therefore not a rational number....
1.6111149258083767361111... .
Like tetration
Tetration
In mathematics, tetration is an iterated exponential and is the next hyper operator after exponentiation. The word tetration was coined by English mathematician Reuben Louis Goodstein from tetra- and iteration. Tetration is used for the notation of very large numbers...
, there is currently no accepted method of extension of the exponential factorial function to real
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
and complex
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
values of its argument, unlike the factorial
Factorial
In mathematics, the factorial of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n...
function, for which such an extension is provided by the gamma function
Gamma function
In mathematics, the gamma function is an extension of the factorial function, with its argument shifted down by 1, to real and complex numbers...
.