The number of cascade functions

Cascade function is a Boolean function which can be implemented by a so-called cascade network. An n input cascade network is a circuit built with n-1 two-input-one-output gates (i.e., dyadic operations) such that at least one input of each gate is a network input. By arranging the inputs in a proper order these networks can be presented in a "cascade" shape, which is the origin of the name. Cascade networks, and thus cascade functions have many interesting properties. Although the portion of cascades among all logic functions is small (the ratio approaches zero with growth of the number of variables), their remarkable properties and the practical significance of such networks has attracted many researchers in the fields of switching functions and logic design. There are several papers published that focus on the enumeration problem of cascade functions and networks. Asymptotic expression and a recurrence relation have been found for the number of all cascades, as well as for some subclasses. However, to the author's knowledge, until now no closed formula has been discovered which explicitly counts such functions. This paper presents an explicit formula for the number of all n-variable cascade functions.