My independent discovery of the sequence A007526

I came across a rather surprising property of the above recurrence relation which I defined when counting the number of models of conditional logics with Lewis-Stanlaker type semantics, i.e. similarity spheres. Because the structure of nested sphere assignments happens to correspond in its form to permutations of non-empty subsets of some set, the formula, known to combinatorialists since Bernoulli, had emerged.

Below is an example taken from The Encyclopedia of Integer Sequences: sequence A007526 entry.

I say those properties were surprising given the context, for in and of themselves they’re not surprising at all. I simply didn’t expect such interesting properties to be inherent in the structure of the systems I was studying.

The sequence s is related to the number of models, given certain parameters (more precisely, the parameter is the size of the filter which is the set of all subformulae of a formula whose satisfiability is being tested - hence the requirement to generate all and only those models which are sufficient and necessary for that task), and given by the following recursive definition:

And this is the "curious" result, which links the above recursively defined function with Euler's Constant e:

It's easy to show that the explicit solution to the above recursive definition of s is given by the formula:

And considering the well known Taylor series, it's easy to show that the result stated above follows.

It turns out that this is a known sequence A007526., whose earliest references are Izquierdo (1659), Caramuel de Lobkowitz (1670), Prestet (1675) and Bernoulli (1713). So it appears that I've discovered/defined this sequence independently of those great mathematicians.