I have read many questions that ask for a closed-form formula for the function $f$ that takes a natural number $n$ and outputs the number of associative operations on an $n$-element set. So far, we only know upper and lower bounds for $f$. I suspect now that there is in fact no closed-form formula. So, my question is, has it been proven that there is no closed-form formula? Of course, to answer this question, we need to know a rigorous definition of a closed-form formula for a function. Has any paper or text rigorously defined closed-form formula, and proven that there is no closed-form formula?
2026-03-30 13:59:16.1774879156
Is there even a closed-form formula for the number of associative operations on an $n$-element set?
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