A binomial double sum I was doing some numerical experiment and I found that
$$\sum_{k=2}^{\infty}\frac{1}{k(k-1)}\sum_{n=k}^{\infty}\frac{1}{\binom{n}{k}(n+1)}=\sum_{k=2}^{\infty}\frac{1}{(k-1)k^2}=2-\zeta(2).$$
I wonder if the following variation
$$\sum_{k=2}^{\infty}\frac{1}{k(k-1)}\sum_{n=k}^{\infty}\frac{1}{\binom{n}{k}(n-1)}$$
has a closed form. Is it related to $\zeta(2)$ or $\zeta(3)$?
 A: We have that the inner sum equals
$$
\eqalign{
  & \sum\limits_{k\, \le \,n} {{1 \over {\binom{n}{k}\left( {n - 1} \right)}}}  =   \cr 
  &  = \sum\limits_{0\, \le \,n} {{1 \over {\binom{n+k}{k}\left( {n + k - 1} \right)}}}  =   \cr 
  &  = k!\sum\limits_{0\, \le \,n} {{1 \over {\left( {n + k} \right)^{\,\underline {\,k\,} } \left( {n + k - 1} \right)}}}  =   \cr 
  &  = k!\sum\limits_{0\, \le \,n} {{1 \over {\left( {n + 1} \right)^{\,\overline {\,k\,} } \left( {n + k - 1} \right)}}}  \cr} 
$$
Therefore
$$
\eqalign{
  & S = \sum\limits_{2\, \le \,k} {{1 \over {k\left( {k - 1} \right)}}\sum\limits_{k\, \le \,n} {{1 \over {\binom{n}{k}\left( {n - 1} \right)}}} }  =   \cr 
  &  = \sum\limits_{2\, \le \,k} {{{k!} \over {k\left( {k - 1} \right)}}\sum\limits_{0\, \le \,n}
 {{1 \over {\left( {n + 1} \right)^{\,\overline {\,k\,} } \left( {n + k - 1} \right)}}} }  =   \cr 
  &  = \sum\limits_{0\, \le \,k} {{{\left( {k + 2} \right)!} \over {\left( {k + 2} \right)\left( {k + 1} \right)}}\sum\limits_{0\, \le \,n}
 {{1 \over {\left( {n + 1} \right)^{\,\overline {\,k + 2\,} } \left( {n + 1 + k} \right)}}} }  =   \cr 
  &  = \sum\limits_{0\, \le \,n} {\sum\limits_{0\, \le \,k} {{{k!} \over {\left( {n + 1} \right)^{\,\overline {\,k + 2\,} } \left( {n + 1 + k} \right)}}} }  =   \cr 
  &  = \sum\limits_{0\, \le \,n} {\sum\limits_{0\, \le \,k} {{{k!\,\left( {n + 1} \right)^{\,\overline {\,k\,} } }
 \over {\left( {n + 1} \right)^{\,\overline {\,k + 2\,} } \left( {n + 1} \right)^{\,\overline {\,k + 1\,} } }}} }  =   \cr 
  &  = \sum\limits_{0\, \le \,n} {{1 \over {\left( {n + 1} \right)^{\,2} \left( {n + 2} \right)}}\sum\limits_{0\, \le \,k} 
 {{{1^{\,\overline {\,k\,} } \,\left( {n + 1} \right)^{\,\overline {\,k\,} } } \over {\left( {n + 3} \right)^{\,\overline {\,k\,} } \left( {n + 2} \right)^{\,\overline {\,k\,} } }}} }  =   \cr
  &  = \sum\limits_{0\, \le \,n} {{1 \over {\left( {n + 1} \right)^{\,2} \left( {n + 2} \right)}}
  {}_3F_{\,2}\left( {\left. {\matrix{  {1,\;1,n + 1}  \cr    {n + 3,n + 2}  \cr  } \;} \right|\;1} \right)}  \cr} 
$$
Unfortunately, the Hypergeometric does not look to be
of a type expressible in a simpler way.
