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| An '''exponential factorial''' is a positive integer ''n'' [[exponentiation|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,
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| : <math>n^{(n - 1)^{(n - 2) \cdots }}.\,</math>
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| The exponential factorial can also be defined with the [[recurrence relation]]
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| : <math>a_0 = 1,\quad a_n = n^{a_{n - 1}}.\,</math>
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| The first few exponential factorials are [[1 (number)|1]], [[1 (number)|1]], [[2 (number)|2]], [[9 (number)|9]], 262144, etc. {{OEIS|id=A049384}}. So, for example, 262144 is an exponential factorial since
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| : <math>262144 = 4^{3^{2^{1}}}.\,</math>
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| The exponential factorials grow much more quickly than regular [[factorial]]s or even [[Factorial#Hyperfactorial|hyperfactorial]]s. The exponential factorial of 5 is 5<sup>262144</sup> which is approximately 6.206069878660874 × 10<sup>183230</sup>.
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| The sum of the reciprocals of the exponential factorials from 1 onwards is the [[transcendental number]] 1.6111149258083767361111... {{OEIS2C|id=A080219}}.
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| Like [[tetration]], there is currently no accepted method of extension of the exponential factorial function to [[real number|real]] and [[complex number|complex]] values of its argument, unlike the [[factorial]] function, for which such an extension is provided by the [[gamma function]].
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| ==References==
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| *Jonathan Sondow, "[http://mathworld.wolfram.com/ExponentialFactorial.html Exponential Factorial]" From [[Mathworld]], a Wolfram Web resource
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| [[Category:Factorial and binomial topics]]
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| [[Category:Integer sequences]]
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| [[Category:Large integers]]
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| {{Numtheory-stub}}
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Emilia Shryock is my title but you can contact me something you like. California is our beginning place. Bookkeeping is my profession. One of the things he loves most is ice skating but he is struggling to find time for it.
Stop by my weblog ... mcb-law.net