How to prove closed form for a binomial sum? Let $l\ge 0$ be an integer.
In the process of solving A tough series related with a hypergeometric function with quarter integer parameters we have discovered the following identity:
\begin{eqnarray}
\frac{1}{4} \sum\limits_{j=0}^{l-1} \sum\limits_{\begin{array}{r} p=-l-1\\p\neq -1 \end{array}}^{l-1}
\binom{-3/2}{j} \binom{-1/2}{l-1-j} \frac{\binom{2j+1}{p+j+1} - \binom{2j+1}{p+j+2}}{2^{2 j+1}}\cdot \frac{(-1)^{p+l}}{p+1}\cdot \left( 1+(-1)^p\right)=\frac{(1/2)^{(l)}}{l!}-1
\end{eqnarray}
Is it possible to prove that identity in some other way, i.e. without resorting to the calculations in the aforementioned question?
 A: This question is actually easier than I thought and it can be solved using elementary methods. First of all note that :
\begin{equation}
\binom{2j+1}{p+j+1}- \binom{2j+1}{p+j+2}= \binom{2j+1}{p+j+1} \left(1-\frac{j-p}{p+j+2}\right)= \binom{2j+1}{p+j+1}\cdot \frac{2p+2}{p+j+2}
\end{equation}
Therefore the sum over $p$ gives:
\begin{eqnarray}
\sum\limits_{p=-l-1,p\neq-1}^{l-1} \frac{\binom{2j+1}{p+j+1}-\binom{2j+1}{p+j+2}}{2^{2j+1}}\cdot \frac{(-1)^{p+l}}{p+1} \cdot (1+(-1)^p)=\\
\sum\limits_{p=-l-1,p\neq-1}^{l-1} \frac{\binom{2j+1}{p+j+1}}{2^{2j+1}}\cdot \frac{2p+2}{p+j+2}\cdot \frac{(-1)^{p+l}}{p+1} \cdot (1+(-1)^p)=\\
\frac{1}{2^{2 j}}\cdot \frac{(-1)^l}{2j+2}\sum\limits_{p=-l-1}^{l-1} \binom{2j+2}{p+j+2} \cdot (1+(-1)^p)=\\
\frac{1}{2^{2 j}}\cdot \frac{(-1)^l}{2j+2}\sum\limits_{p=0}^{j+l+1}
\binom{2j+2}{p} \cdot (1+(-1)^{p+j})= \frac{(-1)^l}{j+1} \cdot 2
\end{eqnarray}
Therefore the sum in question reads:
\begin{eqnarray}
lhs&=& \frac{1}{4} \sum\limits_{j=0}^{l-1} \binom{-3/2}{j} \binom{-1/2}{l-1-j} \cdot \frac{(-1)^l}{j+1} \cdot 2\\
&=&-\sum\limits_{j=1}^l \binom{-1/2}{j} \binom{-1/2}{l-j} \cdot (-1)^l\\
&=&-1+\sqrt{\pi}(-1)^l \frac{1}{l! (-1/2+l)!}\\
&=&-1+\frac{(l-1/2)!}{l!(-1/2)!}\\
&=&-1+\frac{(1/2)^{(l)}}{l!}
\end{eqnarray}
