In H. S. Wilf's generatingfunctionology, (1.6.8) describes:
$$ A_n(y) = \sum_k \begin{Bmatrix}n-1\\k-1\end{Bmatrix} y^k + \sum \begin{Bmatrix}n-1\\k\end{Bmatrix} y^k $$ $$ = yA_{n-1}(y) + \left( y \frac{d}{dy} \right) A_{n-1}(y) $$
where
$$ A_n(y) = \sum_k \begin{Bmatrix}n\\k\end{Bmatrix} y^k $$
but surely if
$$ A_{n-1}(y) = \sum_k \begin{Bmatrix}n-1\\k\end{Bmatrix} y^k = \begin{Bmatrix}n-1\\1\end{Bmatrix}y + \ldots + \begin{Bmatrix}n-1\\k\end{Bmatrix}y^k $$ then $$ A_n(y) = \left (yA_{n-1}(y) - \begin{Bmatrix}n-1\\k\end{Bmatrix} y^{k+1} \right)+ y \frac{d}{dy} A_{n-1}(y) $$
since $A_n(y)$ only sums over $k$ rather than $k+1$. So is this a typo or am I missing something? This is also somewhat important as the author uses this as an example further on to describe other important things.
(I also can't seem to find the errata for the book).
If I understand correctly, it seems like you've confused what the author means by $\sum_k f(k)$. That is, $$\sum_k f(k) \neq f(1) + f(2) + \cdots + f(k)$$ Rather: $$\sum_k f(k) = \sum_{k\in \mathbb{Z}} f(k) = f(0) + f(1) + f(-1) + f(2) + f(-2) + \cdots$$
(See Convention 1.3.)
Taking into account that $\begin{Bmatrix}n\\k\end{Bmatrix} = 0$ if $k > n$ or $n,k<0$, we can see that: \begin{align} A_n(y) &= \sum_k \begin{Bmatrix}n\\k\end{Bmatrix}y^k = \begin{Bmatrix}n\\0\end{Bmatrix} + \begin{Bmatrix}n\\1\end{Bmatrix}y +\cdots + \begin{Bmatrix}n\\n\end{Bmatrix}y^n\\ \implies A_{n-1}(y) &= \sum_k \begin{Bmatrix}n-1\\k\end{Bmatrix}y^k = \begin{Bmatrix}n-1\\0\end{Bmatrix} + \begin{Bmatrix}n-1\\1\end{Bmatrix}y +\cdots + \begin{Bmatrix}n-1\\n-1\end{Bmatrix}y^{n-1}\\ \end{align}
With this, the formulas become readily apparent.