[Bell #s Generalized: e^e^..^(e^x-1)]

[Christian G. Bower]

Leroy Quet <qqquet at mindspring.com> wrote:
> It seems like I have heard of this before. And the idea is obvious.
> But the next-highest-order sequence above the standard Bell numbers is 
> not in the on-line Encyclopedia of Integer Sequences, IF I have 
> calculated the first few terms correctly (by hand).
...
> By the way, {B(3,k)} to start:
> 
> 1, 1, 2, 6, 23, 106,...
> 
> Am I even right about any of this? 
> (If not, this would explain why {B(3,k)} is not in the EIS.)

I get
B(2,k) A000110
B(3,k) A000258
B(4,k) A000307
B(5,k) A000357
B(6,k) A000405
B(7,k) A001669
B(8,k) A081624
B(9,k) A081629

I don't know what happened in the calculation.
I at least expect that B(n,2)=n which is not the case for your
calculation.