I find myself needing to compute (or asymptotically estimate) the following sum over the $2^{S-1}$ compositions of $S$. I am hoping an expert in combinatorics (I am a computer scientist) will recognise this summation.

Let $B_{S,k}$ denote the set of compositions that have exactly $k$ parts, and define $T_{k}$ as,

$$T_{k}=\sum_{a \in B_{S,k}}\frac{a_{1}^{a_{2}}a_{2}^{a_{3}}...a_{k-1}^{a_{k}}}{a_{1}!a_{2}!...a_{k}!},\:\: \text{with}\:\: T_{1}=1.$$

The sum that I am interested in is $\sum_{k}T_{k}$.

Does an expression in terms of $S$ (either exact, or asymptotic) exist, or can it be found?