We know that the recurrence relation for the Towers of Hanoi problem is: $\begin{cases} t_0 = 0; \\ T_n = 2T_{n-1}+1, \ \ \text{for} \ n >0 \end{cases}$. Also we know that $T_n = 2^n-1$.
But is finding the generating function $A(x)$ significant at all? E.g. $A(x) = \sum_{n \geq 0} a_nx^n$ where the $a_n$'s are the $n$th terms of $T_n$? Here $A(x) = \frac{x}{(1-x)(1-2x)}$.
Well this derives the explicit formula for $T_n$. However, is there any special significance of using generating functions to look at the Towers of Hanoi problem?