I would like to have an analytical expression for the Cholesky decomposition of the following matrix: \begin{equation} \mathbf A = \left [ \begin{array}{cccc} 1/1 & 1/2 & 1/3 & 1/4 \\ 1/2 & 1/3 & 1/4 & 1/5 \\ 1/3 & 1/4 & 1/5 & 1/6 \\ 1/4 & 1/5 & 1/6 & 1/7 \\ \end{array} \right ]. \end{equation}
Really, I would be interested in any proof that it is positive definite.