Hanoi graphs and some classical numbers

The Hanoi graphs Hpn model the p-pegs n-discs Tower of Hanoi problem(s). It was previously known that Stirling numbers of the second kind and Stern's diatomic sequence appear naturally in the graphs Hpn. In this note, second-order Eulerian numbers and Lah numbers are added to this list. Considering a variant of the p-pegs n-discs problem, Catalan numbers are also encountered.