User blog comment:Googleaarex/Extended HyperNested Arrays/@comment-5150073-20130407201559

If you findings are all correct, then I believe:

[1 \_(1,4) 2] => theta(e_(W+1)^2)

[1 \_(1,5) 2] => theta(e_(W+1)^3)

[1 \_(1,1,2) 2] => theta(e_(W+1)^w)

[1 \_(1,1,1,2) 2] => theta(e_(W+1)^(w^2))

[1 \_(1 [2] 2) 2] => theta(e_(W+1)^(w^w))

[1 \_(1 [3] 2) 2] => theta(e_(W+1)^(w^(w^2)))

[1 \_(1 [1,2] 2) 2] => theta(e_(W+1)^(w^(w^w)))

Now let [1 \_(X) 2] = [X]_2. Continuing onwards:

[1 \ 2]_2 => theta(e_(W+1)^e_0)

[1 \ 1 \ 2]_2 => theta(e_(W+1)^z_0)

[1 \\ 2]_2 => theta(e_(W+1)^theta(w))

[1 \\\ 2]_2 => theta(e_(W+1)^theta(w^2))

[1 [1 --| 3] 2]_2 => theta(e_(W+1)^theta(W))

[1 [1 [1 \_(3) 3] 2] 2]_2 => theta(e_(W+1)^theta(W^W^W))

[1 \_(1,2) 2]_2 => theta(e_(W+1)^theta(e_(W+1))

Then [1 \_(X) 2]_(Y) = [X]_(Y+1).

[1 \ 2]_3 => theta(e_(W+1)^theta(e_(W+1)^e_0))

[1 \ 2]_4 => theta(e_(W+1)^theta(e_(W+1)^theta(e_(W+1)^e_0)))

[1 \ 2]_(1,2) => theta(e_(W+1)^W)