In https://arxiv.org/abs/1412.1505, the section "Results on Data Complexity" mentions the fact that since the authors are about to proove $\#P_1$ complexity for weighted model counting in Fist Order Logic. Hence, such a problem cannot admit a closed form formula.
Why is this claim true ? Are there counting problems which are hard in general but admit a closed form counting formula ?
- FO$^2$ admits model counting in polynomial time with respect to the domain elements, and also admits a closed form formula for counting.