User blog comment:Lord Aspect/Aspectplex/@comment-32783837-20180315184309

Pro tip: Don't just use "!!!...!!!", that tends to denote the multifactorial, not nested factorials. Use !n instead.

Anyway, if we interpret "!!!...!!!" as nested factorials just for this instance (like you seem to imply),

a(n) = 3!Tn where Tn is the nth triangular number.

The nth triangular number can be computed as \(\frac{n \times n+1}{2}\), so T(200) = 20,100.

However, you've added an extra factorial, so α = 3!20,101.

This is approximately E(1749.6572163922276)#20099.