Transitivity on a set is defined as follows $$\forall x \forall y\forall z( T(x,y) \land T(y,z) \rightarrow T(x,z))$$ Now if we wanted to count total number transitive relations which are defined on an arbitrary set, then we don't have a closed-form expression to do so. The sequence is given here https://oeis.org/A006905

My question is whether you know of any non-trivial expression, which requires at least three variables and one binary predicate and only uses universal quantifiers to be expressed (i.e requires $\mathrm{FO}_3$ with one binary predicate and only universal quantifiers), which has a closed-form counting expression. i.e a generator function that takes the cardinality of domain elements and maps them to the model count of the $\mathrm{FO}_3$ expression.

Through this investigation, I want to understand if there is a link between the countability of a certain process with its expressibility in first-order logic.

Example of what it means that a closed-form counting expression exists. $$\forall_x\forall_y \mathsf{Smokes}(x) \land \mathsf{Friend}(x,y) \rightarrow S(y)$$ If we have $n$ individuals, then the number of models of this expression is given below. Notice this entire expression is expressed in $\mathrm{FO}_2$ $$a(n) = (3^{n}+4^{n})^{n}$$ Source: https://www.ijcai.org/Proceedings/16/Papers/607.pdf