Finding and estimating the number of repetition-free Boolean functions over the elementary basis in the form of a convergent series

We obtain a representation of the number $K_n$ of repetition-free Boolean functions of $n$ variables over the elementary basis $\{\&,\vee,\bar{}\,\}$ in the form of a convergent exponential power series. This representation is the simplest representation among a number of similar formulas containing different combinatorial numbers. The obtained result gives a possibility to find the asymptotics of $K_n$.