Number of Standard Young Tableaux of $n$ cells

Question

I know there is a $1-1$ correspondence between the number of standard young tableaux of $n$ cells and the number of involutions in $S_n$. Number of involutions in $S_n$ satisfies the recurrence relation \begin{equation} a_{n+1}=a_n+na_{n-1}\end{equation} How can we prove that the number of standard young tableaux of $n$ cells also satisfies this relation without using the correspondence with the number of involutions?