Exponential factorial

An exponential factorial is a positive integer n raised to the power of n - 1, which in turn was raised to the power of n - 2, and so on and so forth, that is, Failed to parse (Missing texvc executable; please see math/README to configure.): n^{(n - 1)^{(n - 2) \dots }} . The exponential factorial can also be defined with the recurrence relation Failed to parse (Missing texvc executable; please see math/README to configure.): a_0 = 0, a_n = n^{a_{n - 1}} .

The first few exponential factorials are 0, 1, 2, 9, 262144, etc. (sequence A049384 in OEIS). So, for example, 262144 is an exponential factorial since Failed to parse (Missing texvc executable; please see math/README to configure.): 262144 = 4^{3^{2^{1^0}}} . The exponential factorials grow much more quickly than regular factorials or even hyperfactorials. The exponential factorial of 5 is approximately 6.206069878660874 × 10¹⁸³²³⁰, which is Failed to parse (Missing texvc executable; please see math/README to configure.): 5^{262144} .

The sum of the reciprocals of the exponential factorials is the irrational number 1.6111149258083767361111... A080219.