In the XIX century a toolbox of algebraic manipulations that look like nonsense became popular as a starting point for proofs of properties of numerical sequences, particularly the Bernoulli numbers and Euler numbers. This later on became known as the Umbral Calculus.
In this video I show some accessible examples and talk about the effort during the XX century to explain why the Umbral Calculus actually works which in the end revealed a connection with Linear Algebra and turned it into a much more powerful tool in the process.
A submission for 3b1b's Summer of Math Exposition #SoME1
Link to the proof referenced in the video https://proofwiki.org/wiki/Sum_over_k_of_r_Choose_k_by_s%2Bk_Choose_n_by_-1%5Er-k#Corollary