There seems to be no response to this, but perhaps somebody knows something about it in another terminology.
Franklin T. Adams-Watters defined a triangle similar to Pascal's, but where the latter has c = a + b between and under a and b, Adams-Watters takes c = (a+b)/gcd(a,b). The first few rows look like this:
1 1 1 1 2 1 1 3 3 1 1 4 2 4 1 1 5 3 3 5 1 1 6 8 2 8 6 1 1 7 7 5 5 7 7 1 1 8 2 12 2 12 2 8 1
Adams-Watters summed the rows and obtained a sequence a(n) which is numbered A125606 in the OEIS database. He conjectures that log(a(n))/n tends to 2zeta(3)/zeta(2)log(2*zeta(3)/zeta(2)). That was what I meant when I asked about asymptotic information. (The analogous limit is log(2) for Pascal's triangle, of course)
I am actually more interested in whether the Adams-Watters triangle has any nontrivial arithmetic structure. When you add a and b, the sum is trivially divisible by gcd(a,b), so the operation (a+b)/gcd(a,b) removes this trivial information. Pascal's triangle has a lot of arithmetic structure, but I did not see any when I factored the entries in the A-W triangle out to the fiftieth row. Perhaps dividing by the gcd leaves nothing, or maybe I didn't think about it the right way.
Two caveats: The little piece above is not enough to see what happens. For example, the twelfth row is the first that has both odd and even entries (apart from the 1s). Also there seems no reason to think that the A-W triangle has any combinatorial significance.