If I haven't made a mistake then the first few values of your sequence look like this:

 
$$ (a_{m,n})=\begin{pmatrix}  1 & 1 & 1 & 1 & 1 & 1 & ... \newline
                             1 & 1 & 1 & 1 & 1 & 1 & ... \newline
                             1 & 2 & 3 & 4 & 5 & ... & \newline
                             1 & 3 & 9 & 21 & ... & & \newline
                             1 & 4 & 39/2 & ... & & & \newline
                             1 & 5 & \vdots &  & & & \newline
                            \vdots & \vdots & & & & & \end{pmatrix} $$

In other words, (again provided that I didn't miscalculate anything) your assertion is unfortunately not correct.