Exponential Factorial

The exponential factorial is defined by the recurrence relation

(1)

where . The first few terms are therefore

			(2)
			(3)
			(4)
			(5)

... (OEIS A049384). The term  has  digits.

The exponential factorial is therefore a kind of "factorial power tower."

The sum of the reciprocals of the exponential factorials is given by

			(6)
			(7)

(OEIS A080219). This sum is a Liouville number and is therefore transcendental.

REFERENCES:

Sloane, N. J. A. Sequences A049384 and A080219 in "The On-Line Encyclopedia of Integer Sequences."