Let the top sequence be the square numbers \[h_1(x)=\frac{1}{x+\frac{4}{x+\frac{9}{x+\frac{16}{x+\cdots}}}}\] we find that \[h_1(x)=\frac{1}{x}-\frac{4}{x^3}+\frac{52}{x^5}-\frac{1251 \pm 1}{x^7}\cdots\] this does not seem to match any sequences in OEIS. Although the first \(3\) terms do, there are no sequences with a fourth term close to \(1250\). Let the top sequence be the square root numbers \[h_2(x)=\frac{1}{x+\frac{\sqrt{2}}{x+\frac{\sqrt{3}}{x+\frac{2}{x+\cdots}}}}\] we find that the coefficients are non-integral\[h_2(x)=\frac{1}{x}-\frac{\sqrt{2}}{x^3}+\frac{2+\sqrt{6}}{x^5}-\frac{\approx18.9}{x^7}\cdots\] The term \(\sqrt{2}\) is fairly certain in the above, \(2+\sqrt{6}\) may be incorrect, but it is plausible from the converged result. Not more guesses were made after this. This points to a potential property that integer sequences give integer coefficients. Next we may have the sequence \[h_3(x)=\frac{1}{x+\frac{1}{x+\frac{1}{x+\frac{1}{x+\cdots}}}}\] where all the top row terms are \(1\). This means when \(x=1\), we should have \(h_3(1)=\phi\), the golden ratio. We end up with a few more predictable/obvious terms giving \[h_3(x)=\frac{1}{x}-\frac{1}{x^3}+\frac{2}{x^5}-\frac{5}{x^7}+\frac{14}{x^9}-\frac{42?}{x^{11}}+\frac{132?}{x^{13}}-\cdots\] this suggests the Catalan Numbers \(C(n)=A000108\). In the OEIS, it is listed that the Hankel Transform of these numbers is the sequence \(1,1,1,1,1...\) therefore, it may be that this is the inverse Hankel Transform.
Although this is ill defined (\(x=1\) never converging) to an extent, it would also indicate that \[\sum_{n=1}^\infty (-1)^{n-1}C(n) = \phi\] which is perplexing, as how could the sum of integers only lead to an/(the most) irrational number. However, it could be in a regularised sense, in the same way that \[\sum_{n=1}^\infty n = \frac{-1}{12}\]
Transforming the Catalan Numbers back again in \[c(x)=\frac{1}{x+\frac{1}{x+\frac{2}{x+\frac{5}{x+\cdots}}}}\] gives \[c(x)= \frac{1}{x} - \frac{1}{x^3} + \frac{3}{x^5} - \frac{19}{x^7} + \frac{\approx278}{x^9} -\cdots\] which indicates the transform is not it’s own inverse.
We can see that there is merely an interesting overlap with the Hankel Transform that was described above. It does not correspond to all sequences so far discovered, also, it is noted that the Hankel transform is many to one mapping in [Layman, J. W. “The Hankel Transform and Some of Its Properties.” J. Integer Sequences 4, No. 01.1.5, 2001.].