A closed form for $\sum_{k=0}^n \frac{ (-1)^k {n \choose k}^2}{k+1}$ Mathematica gives  $$\sum_{k=0}^n \frac{(-1)^k {n \choose k}^2}{k+1}= ~_2F_1[-n,-n;2;-1],$$ where $~_2F_1$ that is Gauss hypergeometric function. Here the question is: Can one find a simpler closed form for this summation. Recently, the absolute summation for this has been discussed at MSE:
A binomial summation: $\sum_{k=0}^{n} \frac{{n \choose k}^2}{k+1}$
 A: Use Binomial identity:
$$(1+t)^n=\sum_{k=0}^{n} {n \choose k}t^n~~~(1)$$
Integration of (1) from $t=0$ to $t=x$,gives
$$\frac{(1+x)^{n+1}-1}{n+1}= \sum_{k=0}^n {n \choose k}\frac{x^{k+1}}{k+1}~~~(2)$$
Let $t=-1/x$ in (1), then
$$(-1)^n x^{-n} (1-x)^n=\sum_{k=0}^{n} (-1)^k {n \choose k} x^{-k}~~~~(3)$$
Multiplying (2) and (3) and collecting the terms of $x^1$, we get
$$\frac{(-1)^n}{n+1} [(1-x^2)^{n}(1+x)-(1-x)^n]= x^n\sum_{k=0}^{n} \frac{(-1)^k {n \choose k}^2}{k+1} x^1+...+...$$
$$\implies S_n=\sum_{k=0}^n \frac{(-1)^k {n \choose k}^2}{k+1}=[x^{n+1}] \left((-1)^n \frac{ (1-x^2)^{n}(1+x)-(1-x)^n}{n+1}\right)$$ 
if $m=n/2]$, then
$$S_n=(-1)^{m} \frac{{n \choose m}}{n+1}.$$
A: We seek to evaluate
$$\sum_{k=0}^n \frac{(-1)^k}{k+1} {n\choose k}^2.$$
This is
$$\frac{1}{n+1} \sum_{k=0}^n (-1)^{k} {n\choose k} {n+1\choose k+1}
= \frac{1}{n+1} \sum_{k=0}^n (-1)^{k} {n\choose k} {n+1\choose n-k}
\\ = [z^n] (1+z)^{n+1} \frac{1}{n+1}
\sum_{k=0}^n (-1)^{k} {n\choose k} z^k
\\ = [z^n] (1+z)^{n+1} \frac{1}{n+1} (1-z)^n
= \frac{1}{n+1} [z^n] (1+z) (1-z^2)^n.$$
Now if $n=2m$ we get
$$\frac{1}{n+1} [z^{2m}] (1+z) (1-z^2)^n
= \frac{1}{n+1} [z^{2m}] (1-z^2)^n
\\ = \frac{1}{n+1} [z^{m}] (1-z)^n
= \frac{1}{n+1} (-1)^m {n\choose m}.$$
On the other hand when $n=2m+1$ we find
$$\frac{1}{n+1} [z^{2m+1}] (1+z) (1-z^2)^n
= \frac{1}{n+1} [z^{2m+1}]z  (1-z^2)^n
\\ = \frac{1}{n+1} [z^{2m}] (1-z^2)^n
= \frac{1}{n+1} [z^{m}] (1-z)^n
= \frac{1}{n+1} (-1)^m {n\choose m}.$$
We thus have even or odd the closed form
$$\bbox[5px,border:2px solid #00A000]{
\frac{1}{n+1} (-1)^{\lfloor n/2\rfloor}
{n\choose \lfloor n/2\rfloor}.}$$
The second case could have been done by inspection given the first.
This result matches the comment by @SangchulLee.
A: $\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,}
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 \newcommand{\mrm}[1]{\mathrm{#1}}
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\begin{align}
\sum_{k = 0}^{n}{\pars{-1}^{k}{n \choose k}^{2} \over k + 1} & =
\int_{0}^{1}\sum_{k = 0}^{n}{n \choose k}^{2}\pars{-t}^{k}\,\dd t =
\int_{0}^{1}\sum_{k = 0}^{n}{n \choose k}\pars{-t}^{k}
\bracks{z^{n - k}}\pars{1 + z}^{n}\,\dd t
\\[5mm] & =
\bracks{z^{n}}\pars{1 + z}^{n}\int_{0}^{1}\sum_{k = 0}^{n}
{n \choose k}\pars{-tz}^{k}\,\dd t
\\[5mm] & =
\bracks{z^{n}}\pars{1 + z}^{n}\int_{0}^{1}\pars{1 - tz}^{n}\,\dd t =
\bracks{z^{n}}\pars{1 + z}^{n}\,
{\pars{1 - z}^{n + 1} - 1 \over -\pars{n + 1}z}
\\[5mm] & =
-\,{1 \over n + 1}\bracks{z^{n + 1}}\pars{1 - z^{2}}^{n}\pars{1 - z} 
\\[5mm] & =
{1 \over n + 1}\braces{\bracks{z^{n}}\pars{1 - z^{2}}^{n} -
\bracks{z^{n + 1}}\pars{1 - z^{2}}^{n}}
\\[5mm] & =
\bbx{\left\{\begin{array}{lcl}
\ds{{1 \over n + 1}{n \choose n/2}\pars{-1}^{n/2}} &
\mbox{if} & \ds{n}\ \mbox{is}\ even
\\[3mm]
\ds{{1 \over n + 1}{n \choose \bracks{n + 1}/2}\pars{-1}^{\pars{n - 1}/2}} &
\mbox{if} & \ds{n}\ \mbox{is}\ odd
\end{array}\right.} \\ &
\end{align}
A: One way to look at this is to take:
$\begin{align*}
  \sum_k \frac{(-1)^k}{k + 1} \binom{n}{k}^2
    &= \sum_k \frac{(-1)^k}{k + 1} \binom{n}{k} \binom{n}{n - k} \\
    &= [z^n] \left(
                \sum_k \frac{1}{k + 1} \binom{n}{k} z^k
             \right)
               \cdot (-1)^n \left(
                              \sum_k (-1)^k \binom{n}{k} z^k
                            \right)
\end{align*}$
For the pieces, you know that:
$\begin{align*}
   \sum_k \binom{n}{k} z^k
     &= (1 + z)^n \\
   \sum_k \frac{1}{k + 1} \binom{n}{k} z^k
     &= \frac{1}{z} \sum_k \frac{1}{k + 1} \binom{n}{k} z^{k + 1} \\
     &= \frac{1}{z} \int_0^z (1 + t)^n \, d t \\
     &= \frac{1}{z} \frac{(1 + z)^{n + 1} - 1}{n + 1}
\end{align*}$
Thus:
$\begin{align*}
  \sum_k \frac{(-1)^k}{k + 1} \binom{n}{k}^2
    &= (-1)^n [z^n] \frac{(1 + z)^{n + 1} - 1}{z (n + 1)} (1 - z)^n \\
    &= \frac{(-1)^n}{n + 1}
         [z^{n + 1}] ((1 + z)^{n + 1} - 1) (1 - z)^n \\
    &= \frac{(-1)^n}{n + 1}
         [z^{n + 1}] ((1 - z^2)^n (1 + z) \\
    &= \frac{(-1)^n}{n + 1}
         \left(
           [z^n] (1 - z^2)^n
           [z^{n - 1}] (1 - z^2)^n
         \right) \\
    &= \frac{(-1)^n}{n + 1}
         \left(
            (-1)^{n} \binom{2 n}{n}
              + (-1)^{n - 1} \binom{2 n}{n - 1}
         \right) \\
    &= \frac{1}{n + 1} \left( \binom{2 n}{n} - \binom{2 n}{n - 1} \right) \\
    &= \frac{1}{(n + 1)^2} \binom{2 n}{n}
\end{align*}$
