Exponential factorial

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, $n^\{(n\; -\; 1)^\{(n\; -\; 2)\; dots$. The exponential factorial can also be defined with the recurrence relation $a\_0\; =\; 0,\; a\_n\; =\; n^\{a\_\{n\; -\; 1$.

The first few exponential factorials are 0, 1, 2, 9, 262144, etc. OEIS|id=A049384. So, for example, 262144 is an exponential factorial since $262144\; =\; 4^\{3^\{2^\{1^0\}$. The exponential factorials grow much more quickly than regular factorials or even hyperfactorials. The exponential factorial of 5 is $5^\{262144\}$ which is approximately 6.206069878660874 × 10¹⁸³²³⁰.

The sum of the reciprocals of the exponential factorials is the irrational number 1.6111149258083767361111... OEIS2C|id=A080219.

References

* Jonathan Sondow, " [http://mathworld.wolfram.com/ExponentialFactorial.html Exponential Factorial] " From Mathworld, a Wolfram Web resource