Sum of binomial coefficients $\sum_{k=0}^{n}(-1)^k\binom{n}{k}\binom{2n - 2k}{n - 1} = 0$

Question

How do I prove the following identity:

$$\sum_{k=0}^{n}(-1)^k\binom{n}{k}\binom{2n - 2k}{n - 1} = 0$$

I am trying to use inclusion-exclusion, and this will boil down to a sum like inclusion-exclusion, and the $\binom{2n-2k}{n-1}$ term wouldn't matter (it will be equivalent to set sizes). Is this a correct way to go?

Answer 1

In how many ways can you select $m\lt n$ squares on a $2\times n$ board such that exactly $n$ columns contain a selected square?

[Edit:]

From the lack of upvotes and the inquiring comment of a distinguished user I conclude that I should explain this perhaps overly laconic answer.

The OP wanted to prove the result by inclusion–exclusion. The number of ways to select $m$ squares on a $2\times n$ board such that at most $j$ particular columns contain a selected square is $\binom{2j}m$. By inclusion–exclusion, if there are $a_j$ ways to do something with at most $j$ particular objects, then there are

$$ \sum_{k=0}^n(-1)^k\binom nka_{n-k} $$

ways to do it with exactly $n$ objects (where the binomial coefficient counts the number of ways of selecting $n-k$ particular ones of the $n$ objects). Putting this together yields the number of ways to select $m$ squares on a $2\times n$ board such that exactly $n$ columns contain a selected square:

$$ \sum_{k=0}^n(-1)^k\binom nk\binom{2n-2k}m\;. $$

Since it's impossible to have exactly $n$ columns contain a selected square if less than $n$ squares are selected, this is $0$ for $m\lt n$, and thus in particular for $m=n-1$.

Answer 2

The function $g:k\mapsto \binom{2n-2k}{n-1}$ is a polynomial function of degree $n-1$. The operation $f\mapsto\bigl(x\mapsto\sum_{k=0}^n(-1)^k\binom knf(x+k)\bigr)$ equals $(-1)^n\Delta^n$, where $\Delta$ is the finite difference operator $f\mapsto\bigl(x\mapsto f(x+1)-f(x)\bigr)$, which kills constant functions and lowers the degree of polynomial functions by $1$. Therefore $(-1)^n\Delta^n(g)=0$, which means that $$ x\mapsto\sum_{k=0}^n(-1)^k\binom kng(x+k) $$ is the zero function. Now apply for $x=0$ to obtain $$ 0=\sum_{k=0}^n(-1)^k\binom kng(k) = \sum_{k=0}^n(-1)^k\binom kn\binom{2n-2k}{n-1}. $$