# Constructing FOL formula for which counting is easy?

Given a function free First Order Logic language $$\mathcal{L}$$ are there ways to write formulas for which counting the number of models for a given cardinality of the domain is easy (preferably exists a closed-form formula)?

An Example:

Let the language consist of a binary predicate $$T$$ and variables $$\{x,y,z\}$$, and a domain $$\mathcal{D}$$ consisting of $$n$$ elements. Are there ways to write formulas in this language such that counting the number of models for these formulas is computationally easy? A trivial example would be $$\forall x, y,z T(x,x)\land T(y,y) \land T(z,z)$$, for which the number of models would be $$2^{n^2 - n}$$, are there more intricate ways to generate such formulas?