  The goal of this article is to study some basic algebraic and combinatorial
properties of "generalized $n$-series" over a commutative ring $R$, which are
functions $s: \mathbf{Z}_{\geq 0} \to R$ satisfying a mild condition. A special
example of generalized $n$-series is given by the $q$-integers
$\frac{q^n-1}{q-1} \in \mathbf{Z}[\![q-1]\!]$. Given a generalized $n$-series
$s$, one can define $s$-analogues of factorials (via $n!_s = \prod_{i=1}^n
s(n)$) and binomial coefficients. We prove that Pascal's identity, the binomial
identity, Lucas' theorem, and the Vandermonde identity admit $s$-analogues;
each of these specialize to their appropriate $q$-analogue in the case of the
$q$-integer generalized $n$-series. We also study the growth rates of
generalized $n$-series defined over the integers. Finally, we define an
$s$-analogue of the ($q$-)derivative, and prove $s$-analogues of the Poincar\'e
lemma and the Cartier isomorphism for the affine line, as well as a pullback
square due to Bhatt-Lurie.
