$$\text{Find} \ \ \sum_{k=0}^{n} (-1)^k k \binom{n}{k}^2$$
I expanded the binomial coefficients within the sum and got $$\binom{n}{0}^2 + \binom{n}{1}^2 + \binom{n}{2}^2 + \dots + \binom{n}{n}^2$$ What does this equal to? I think this can help me evaluate the original sum.
First, use $k\binom{n\vphantom{1}}{k}=n\binom{n-1}{k-1}=n\binom{n-1}{n-k}$ $$ \sum_{k=0}^n(-1)^kk\binom{n}{k}^2 =n\sum_{k=0}^n(-1)^k\binom{n}{k}\binom{n-1}{n-k}\tag{1} $$ Next compute a generating function. The sum we want is the coefficient of $x^n$ $$ \begin{align} n\sum_{m,k}(-1)^k\binom{n}{k}\binom{n-1}{m-k}x^m &=n\sum_{m,k}(-1)^k\binom{n}{k}\binom{n-1}{m-k}x^{m-k}x^k\\ &=n\sum_k(-1)^k\binom{n}{k}(1+x)^{n-1}x^k\\ &=n(1+x)^{n-1}(1-x)^n\\ &=n\left(1-x^2\right)^{n-1}(1-x)\tag{2} \end{align} $$ The sum we want is the coefficient of $x^n$ in $(2)$: $$ \begin{align} \sum_{k=0}^n(-1)^kk\binom{n}{k}^2 &=\left\{\begin{array}{} n\binom{n-1}{n/2}(-1)^{n/2}&\quad\text{if $n$ is even}\\[6pt] n\binom{n-1}{(n-1)/2}(-1)^{(n+1)/2}&\quad\text{if $n$ is odd} \end{array}\right.\\[6pt] &=n\binom{n-1}{\lfloor n/2\rfloor}(-1)^{\lceil n/2\rceil}\tag{3} \end{align} $$
$$\text{ Hint: }\sum^{n}_{k=0}(-1)^k{n\choose k}^2=(-1)^m{2m\choose m},\ m=\frac{n}{2},\ \forall n: 2\mid n$$ $$\text{and}$$ $$\sum^{n}_{k=0}(-1)^k{n\choose k}^2=0,\ \forall n:2\nmid n$$
This can also be done using a basic complex variables technique. Start as in @robjohn's answer. Suppose we seek to evaluate $$\sum_{k=0}^n (-1)^k k {n\choose k}^2 = \sum_{k=1}^n {n\choose k} (-1)^k k {n\choose k} \\= \sum_{k=1}^n {n\choose k} (-1)^k k \frac{n}{k} {n-1\choose k-1} = n \sum_{k=1}^n {n\choose k} (-1)^k {n-1\choose k-1}.$$
Introduce the integral representation $${n-1\choose k-1} = \frac{1}{2\pi i} \int_{|z|=\epsilon} \frac{1}{z^k} (1+z)^{n-1} \; dz.$$
This gives the following integral for the sum $$\frac{n}{2\pi i} \int_{|z|=\epsilon} \sum_{k=1}^n {n\choose k} (-1)^k \frac{1}{z^k} (1+z)^{n-1} \; dz \\= \frac{n}{2\pi i} \int_{|z|=\epsilon} (1+z)^{n-1} \sum_{k=1}^n {n\choose k} (-1)^k \frac{1}{z^k} \; dz \\= \frac{n}{2\pi i} \int_{|z|=\epsilon} (1+z)^{n-1} \left(-1 + \left(1-\frac{1}{z}\right)^n \right) \; dz$$
We may drop the $-1$ because it participates in a product that is entire. This leaves $$\frac{n}{2\pi i} \int_{|z|=\epsilon} (1+z)^{n-1} \frac{(z-1)^n}{z^n} \; dz \\ = \frac{(-1)^n n}{2\pi i} \int_{|z|=\epsilon} (1-z) (1+z)^{n-1} \frac{(1-z)^{n-1}}{z^n} \; dz \\ = \frac{(-1)^n n}{2\pi i} \int_{|z|=\epsilon} (1-z) \frac{(1-z^2)^{n-1}}{z^n} \; dz.$$
It follows that the value of the sum is given by $$(-1)^n n [z^{n-1}] (1-z) (1-z^2)^{n-1}.$$
For $n$ even the $z$ from $1-z$ participates and for $n$ odd the one participates.
We have for $n$ even the result $$(-1)^n n \times (-1) (-1)^{(n-2)/2} {n-1\choose (n-2)/2} = n \times (-1)^{n/2} {n-1\choose n/2-1} \\= n \times (-1)^{n/2} {n-1\choose n/2}$$ and for $n$ odd $$(-1)^n n \times (-1)^{(n-1)/2} {n-1\choose (n-1)/2} = n \times (-1)^{(n+1)/2} {n-1\choose (n-1)/2}.$$
Joining these two terms we obtain $$n \times (-1)^{\lceil n/2 \rceil} {n-1\choose \lfloor n/2 \rfloor}.$$
A trace as to when this method appeared on MSE and by whom starts at this MSE link.
