I saw a question here, and then I thought,
Let's take $$x_{n;k_1,k_2,k_3\dots k_{n-1}}=\Big(1-\frac{k_1}{3}\Big)^2\Big(1-\frac{k_2}{6}\Big)^2\Big(1-\frac{k_3}{10}\Big)^2\dots \Big(1-\frac{k_{n-1}}{\frac{n(n+1)}{2}}\Big)^2\space n\geq2\\k_1\leq2,k_2\leq 3,\dots k_{n-1}\leq n.$$
What is $$S_n=\sum_{k_1,k_2,k_3,\dots k_{n-1}}x_{n;k_1,k_2,k_3\dots k_{n-1}}$$
or where is it going asymptotically?
So $\Big(1-\frac{k_{m-1}}{\frac{m(m+1)}{2}}\Big)=\frac{1+2+\dots +m-k_{m-1}}{1+2+\dots +m}$. But what next?
