Calculate $\mathrm{det}(P_{n+1})/\mathrm{det}(P_n)$, where $$P_n=\begin{bmatrix} \frac{1}{2}&\frac{1}{3}&\ldots&\frac{1}{n+1}\\ \frac{1}{3}&\frac{1}{4}&\ldots&\frac{1}{n+2}\\ \vdots&\vdots&\ddots&\vdots\\ \frac{1}{n+1}&\frac{1}{n+2}&\ldots&\frac{1}{2n}\\ \end{bmatrix}\,.$$
I have calculated $\mathrm{det}(P_n)$ for small $n$'s: $$ \mathrm{det}(P_1)=\frac{1}{2},\quad \mathrm{det}(P_2)=\frac{1}{72},\quad \mathrm{det}(P_3)=\frac{1}{43200},\quad \mathrm{det}(P_4)=\frac{1}{423360000},$$ which gives $$ \frac{\mathrm{det}(P_2)}{\mathrm{det}(P_1)}=\frac{1}{36},\quad \frac{\mathrm{det}(P_3)}{\mathrm{det}(P_2)}=\frac{1}{600},\quad \frac{\mathrm{det}(P_4)}{\mathrm{det}(P_3)}=\frac{1}{9800}\,. $$ However, I cannot spot any pattern in these small cases. Any ideas? Thanks for reading my question.
