From a symmetry between $e$ and $π$ to the holonomic triangle

In 2002, the author studied together the recurrences $u_{n+2}=u_{n+1}+u_n/n$ and $v_{n+2}=v_{n+1}/n+v_n$, in which the factor $1/n$ is placed in a symmetric way. The first one leads to $e$ and the second one to $π$, and each of them is classical when taken separately. When $n$ is replaced by $n+m$ i