[seqfan] A sequence based on parenthesizations of repeated exponentiation

[Vladimir Reshetnikov]

Hi SeqFans,

There are several sequences in OEIS related to all possible
parenthesizations of repeated exponentiation.

A000081 -- the number of different functions of x that can be obtained by
inserting parentheses in the expression x^x^...^x with n occurrences of x
(assuming the functions are defined for x > 0).

A002845 -- the number of different values that can be obtained by inserting
parentheses in the expression 2^2^...^2 with n occurrences of 2.

Similar sequences for 3, 4, 5, ... are  A003018, A003019, A145545, A145546,
A145547, A145548, A145549, A145550.

Similar sequences for 1/2, sqrt(2), i=sqrt(-1) are A196244, A082499,
A198683.

A199812 -- the number of different ordinals that can be obtained by
inserting parentheses in the expression w^w^...^w with n occurrences of w,
where w is the first infinite ordinal \omega, and ^ denotes the ordinal
exponentiation.

A038055 -- can be considered as a modification of A000081, where each
operand can be either x or y, and we count the number of different
functions of two variables x, y (assuming the functions are defined for x >
0, y > 0)

I propose another sequence of this family, that can be considered a
crossbreed of A002845 and A038055:

a(n) = the number of different functions of x that can be obtained by
inserting parentheses in the expression op^op^...^op with n operands where
each operand op can be either x of 2 (assuming the functions are defined
for x > 0).

It appears that a(n) starts with 2, 4, 10, 36, 140, 564(?), ...
Could you suggest an efficient algorithm to compute more terms?

The first 6 terms look like doubled A014271. Is it indeed the same sequence
(up to a common factor)?

--
Thanks
Vladimir Reshetnikov