Derangements and Continued Fractions for $e$

TL;DR

This paper derives a new elegant continued fraction expansion for the number e, connecting it to derangement numbers and factorial ratios, building on results from the Ramanujan Machine automated conjecture generator.

Contribution

It introduces a novel continued fraction for e based on derangement numbers, providing insight into the structure behind automated conjectures.

Findings

Derived a continued fraction for e using derangement numbers

Connected continued fractions to factorial and subfactorial ratios

Validated the expansion against known results from the Ramanujan Machine

Abstract

Several continued fraction expansions for $e$ have been produced by an automated conjecture generator (ACG) called \emph{The Ramanujan Machine}. Some of these were already known, some have recently been proved and some remain unproven. While an ACG can produce interesting putative results, it gives very limited insight into their significance. In this paper, we derive an elegant continued fraction expansion, equivalent to a result from the Ramanujan Machine, using the sequence of ratios of factorials to subfactorials or derangement numbers.