I'm reading from 3 different sets of notes on generating functions and having a little trouble integrating their approaches.
First, I'm used to working with the following definitions:
$$c(n,k)=\binom{n}{k}=\frac{n!}{k!(n-k)!}$$ and $$D(n,k)=\binom{n-1+k}{k}=\binom{n-1+k}{n-1}$$
One set of notes says for any positive $n,k$ then $$c(-n,k)=(-1)^{k}\binom{n-1+k}{k}$$ gives the $k^{\text th}$ coefficient for $(1+x)^{-n}$
The other 2 sets give me the equation: $$\frac{1}{(1-x)^{n}}=\sum_{k=0}^\infty\binom{n-1+k}{n-1}x^k=\sum_{k=0}^\infty\binom{n-1+k}{k}x^k$$
He then goes on to say this last is the generating function for the series of negative binom coefficients and that it is exactly $D(n,k)$. How can both these be correct? Either the $k^{\text th}$ coefficient is $(-1)^{k}D(n,k)$ or it's $D(n,k)$ but how can it be both?
I'm obviously missing something trivial here I think...