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During two centuries, generating functions were popping up via the corresponding recurrences on their coefficients (as can be seen in the seminal works of [[Bernoulli]]{{dn|date=April 2021}}, [[Leonhard Euler|Euler]], [[Arthur Cayley]], [[Ernst_Schröder|Schröder]],
[[Srinivasa Ramanujan|Ramanujan]], [[John Riordan (mathematician)|Riordan]], [[Donald Knuth|Knuth]],
It was then slowly realized that the generating functions were capturing many other facets of the initial discrete combinatorial objects, and that this could be done in a more direct formal way: The recursive nature of some combinatorial structures
translates, via some isomorphisms, into noteworthy identities on the corresponding generating functions.
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