While answering this question I stumbled on some nice (inexplicable) observation where a recursively defined sequence of polynomials turned out to coincide with some Taylor development
I'll start with the probabilistic interpretation / background
Let $p \in [0; \frac 12]$ and suppose $(X_n)$ is a independant identically distributed random sequence with
$P(X_n = (-1,-1)) = P(X_n = (+1,+1)) = p$ and $P(X_n = (-1,+1)) = 1-2p$.
For $k \in \Bbb Z$ and $n \ge 0$ with $k+n$ even, denote $E_{k,n}$ the event $ \sum_{k=0}^n X_n=(a,b)$ with $a<k<b$.
Then $f_{k,n}(p) = \frac 1 {1-2p} P(E_{k,n})$ is a polynomial in $p$ (in fact it's a polynomial in $p-p^2$), and it turns out that as $n-|k|$ gets larger, they tend to look a lot like $1+2p+4p^2+\ldots$ :
Problem : show that $f_{k,n}(p) = 1/(1-2p) + O(p^{(n-|k|+2)/2})$ (as $p \to 0$)
It can be shown that if we make the generating function $F_p(X,Y) = \sum f_{k,n}(p) X^k Y^n$, then $F_p = (1 - Y(X+X^{-1}) + Y^2(1+q(X-X^{-1})^2))^{-1}$ where $q=p-p^2$.
This allows to compute those polynomials recursively relatively quickly.
The first few terms are
$\begin{array} c
n=0 & & & & & 1 & \\
n=1 & & & & 1 & & 1 \\
n=2 & & & 1-q & & 1+2q & \\
n=3 & & 1-2q & & 1+2q & & 1+2q \\
n=4 & 1-3q+q^2 & & 1+2q-4q^2 & & 1+2q+6q^2 & &
\end{array}$
(the triangle is symmetric so there is no point showing the right half)
$f_{k,n}$ is a polynomial of degree exactly $\lfloor n/2 \rfloor$ in $q$, and making $q$ the indeterminate, the problem has the consequence :
SubProblem :
show that $f_{0,2n}(q)$ and $f_{\pm1,2n+1}(q)$ are the Taylor expansion of $1/\sqrt{1-4q}$ or order $n$
Showing that the new coefficient that appear every two step is the right one $(1,2,6,...)$ is easy, the problem is showing is that the previous coefficients stay the same.
The recurrence relation modulo $q$ or modulo $X-1$ is super nice to show stuff but not quite nice enough to make a recurrence straightforward.
It is known that when $p \in [0;\frac 12]$ and $n \to \infty$, $f_{\lambda n,n}(p)$ converges to $1/(1-2p)$ if $|\lambda| < 1-2p$ and to $0$ for $|\lambda| > 1-2p$
