On p. 49 in Gould's book Combinatorial Identities, the author states that the sum $$\sum\_{k=0}^{n-1}(-1)^k\binom{n}{k}\binom{2n}{2k}^{-1}$$ "... arises naturally in a statistical problem; it amounts to the evaluation of the moments of a certain distribution". Does someone know what exactly Gould is referring to? I am looking for references...



A reference for a sum found in Gould's Combinatorial Identities book
$\renewcommand{\b}{\binom}\renewcommand{\B}{\text{B}}$As noted in [Carlo Beenakker's comment](https://mathoverflow.net/questions/443245/a-reference-for-a-sum-found-in-goulds-combinatorial-identities-book#comment1144176_443245),  

\begin{equation\*}
 s\_n:=\sum\_{k=0}^{n-1}(-1)^k\b nk\b{2n}{2k}^{-1}=\frac{2 n+1-(-1)^n}{2 (n+1)}. \tag{1}\label{1}
\end{equation\*}
The immediately related formula
\begin{equation\*}
\sum\_{k=0}^n(-1)^k\b nk\b{2n}{2k}^{-1}=\frac{[1+(-1)^n](2n+1)}{2 (n+1)}
\end{equation\*}
(with the upper summation limit $n$)
is formula 4.2.8.5 in Vol.1 of the handbook by Prudnikov, Brychkov, and Marichev (in Russian).


It is immediate from \eqref{1} that $(s\_n)$ is not the sequence of moments of any probability distribution. Indeed, otherwise the sequence $(s\_{2n})$ would be convex, which it is not. So, the statement "it amounts to the evaluation of the moments of a certain distribution" cannot be true, unless "amounts to" is understood as something like "has something to do with".


However, it may be of help to present a (quite standard) derivation of \eqref{1}. It is based on the following representation of the reciprocals of the binomial coefficients in terms of the beta function (which of course has to do with the beta distribution):
\begin{equation\*}
 \b sr^{-1}=r\B(r,s-r+1) \tag{2}\label{2}
\end{equation\*}
for integers $r$ and $s$ such that $0<r\le s$,
where $\B(a,b):=\int\_0^1 dx\, x^{a-1}(1-x)^{b-1}$.


By the definition of $s\_n$ in \eqref{1} and identity \eqref{2}, we get
\begin{equation\*}
\begin{aligned} 
s\_n&=1+\sum\_{k=1}^{n-1}(-1)^k\b nk\b{2n}{2k}^{-1} \\
 &=1+\int\_0^1 dx\, \sum\_{k=1}^n(-1)^k\b nk 2k x^{2k-1}(1-x)^{2n-2k} \\ 
 &=1-2n\int\_0^1 dx\,[(-1)^n x^{2n-1}+x(1-2x)^{n-1}] \\ 
 &=\frac{2 n+1-(-1)^n}{2 (n+1)}.\quad\Box
\end{aligned} 
\end{equation\*}
One may note that the latter integral looks something like an expression for the $n$th moment of a probability distribution.


