Original paper:

* Friedmann, T., & Hagen, C. R. (2015). Quantum mechanical derivation of the Wallis formula for π. Journal of Mathematical Physics

https://sci-hub.tw/10.1063/1.4930800

The result didn't come from experimental works, it's a pure theoretical derivation for the sake of mathematical physics.

> The existence of such a derivation indicates that there are striking connections between well-established physics and pure mathematics that are remarkably beautiful yet still to be discovered.

The paper only has three pages - it set up a particular calculation on the energy levels of hydrogen and obtains a limit, thus recovering the Wallis formula for π. If you know QM (I don't), the paper may be a fun read, it's a short and understandable calculation.