I built a Pascal Triangle, but instead of each value in a row being a sum of the two numbers above it I made each value the sum of the numbers above it divided by the product of those numbers. So, as an example the first three lines are 1, 1 1, 1 2 1, but the third line is 1 1.5 1.5 1, and the fourth is 1 1.6 1.35 1.35 1.6 1, etc.
The values appear to be converging (possibly $\sqrt2$?), but I cannot tell for certain if they do as I don't have a grid large enough to continue carrying out the values.
Is there an equation that I can plug in a row number and the element of that row to find what its value would be? I believe that I can use the expression $$n!/k!(n-k)!$$ for Pascal's Triangle, but I don't know how to modify the equation for this variation.