Clarkkkkson

The Clarkkkkson, denoted ¥, is a dynamic googologism that grows over time, based on the lynz.

Let $$n!$$ represent the factorial, and create the following extension:


 * $$n!! = n! \cdot (n - 1)! \cdot (n - 2)! \cdot (n - 3)! \cdot \ldots$$
 * $$n!!! = n!! \cdot (n - 1)!! \cdot (n - 2)!! \cdot (n - 3)!! \cdot \ldots$$
 * $$n!!!! = n!!! \cdot (n - 1)!!! \cdot (n - 2)!!! \cdot (n - 3)!!! \cdot \ldots$$
 * etc.

For the sake of this article, $$n!!$$ will be abbreviated as $$n!2$$, $$n!!!$$ as $$n!3$$, and so forth. The notation is due to Aalbert Torsius.

Define the hypf function as follows (using arrow notation):


 * $$\text{hypf}(c, 2, n) = n!c \cdot (n - 1)!c \cdot (n - 2)!c \cdot \ldots$$
 * $$\text{hypf}(c, 3, n) = n!c \downarrow (n - 1)!c \downarrow (n - 2)!c \downarrow \ldots$$
 * $$\text{hypf}(c, 4, n) = n!c \downarrow\downarrow (n - 1)!c \downarrow\downarrow (n - 2)!c \downarrow\downarrow \ldots$$
 * and so on through the weak hyper-operators.

Further, define the Clarkkkkson function ck(c, p, n, r) = hypf(c, p, hypf(c, p, hypf(c, p, ... n))), where the hypf function is applied r times.

Finally, define the following series:


 * A1 = ck(K, K, K, K) (where K is the lynz)
 * A2 = ck(A1, A1, A1, A1)
 * A3 = ck(A2, A2, A2, A2)
 * etc.
 * The Clarkkkkson is AK + 1.

The Clarkkkkson changes over time. Thus, it can be considered a function of time instead of a true number.

Currently, Clarkkkkson is at most on the level of gigol, and will be for a very long time. The author's use of the weak hyper-operators instead of the strong ones puts ¥ at a significant disadvantage.