Integer sequences | Large integers | Factorial and binomial topics | Exponentials

Exponential factorial

The exponential factorial is a positive integer n raised to the power of n โˆ’ 1, which in turn is raised to the power of n โˆ’ 2, and so on and so forth in a right-grouping manner. 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. (sequence in the OEIS). For example, 262144 is an exponential factorial since Using the recurrence relation, the first exponential factorials are: 111 = 121 = 232 = 949 = 2621445262144 = 6206069878...8212890625 (183231 digits) 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 5ร—10183230. The sum of the reciprocals of the exponential factorials from 1 onwards is the following transcendental number: This sum is transcendental because it is a Liouville number. 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. But it is possible to expand it if it is defined in a strip width of 1. (Wikipedia).

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From playlist Linear Algebra

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Related pages

Transcendental number | Hyperfactorial | Exponentiation | Complex number | Gamma function | Recurrence relation | Factorial | Real number | Liouville number | Tetration