Sum over partitions I want to calculate the following sum over non-negative integer partitions
$$
\sum_{l_1+\cdots +l_n=s} \frac{1}{(l_1!)^2 \cdots (l_n!)^2}.
$$
for fixed $n$ and $s.$
I tried to use Vandermonde's identity and induction, but no success so far. 
 A: $\newcommand{\bbx}[1]{\,\bbox[8px,border:1px groove navy]{{#1}}\,}
 \newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
 \newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
 \newcommand{\dd}{\mathrm{d}}
 \newcommand{\ds}[1]{\displaystyle{#1}}
 \newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,}
 \newcommand{\ic}{\mathrm{i}}
 \newcommand{\mc}[1]{\mathcal{#1}}
 \newcommand{\mrm}[1]{\mathrm{#1}}
 \newcommand{\pars}[1]{\left(\,{#1}\,\right)}
 \newcommand{\partiald}[3][]{\frac{\partial^{#1} #2}{\partial #3^{#1}}}
 \newcommand{\root}[2][]{\,\sqrt[#1]{\,{#2}\,}\,}
 \newcommand{\totald}[3][]{\frac{\mathrm{d}^{#1} #2}{\mathrm{d} #3^{#1}}}
 \newcommand{\verts}[1]{\left\vert\,{#1}\,\right\vert}$
\begin{align}
&\sum_{\ell_{1}\ +\ \cdots\ +\ \ell_{n}\ =\ s}\hspace{3mm}
{1 \over \pars{\ell_{1}!}^{2} \cdots\pars{\ell_{n}!}^{2}} =
\sum_{\ell_{1}\ldots\ell_{n}}\hspace{3mm}
{1 \over \pars{\ell_{1}!}^{2} \cdots\pars{\ell_{n}!}^{2}}
\oint_{\verts{z} = 1^{-}}\,\,\,
{1 \over z^{s + 1 - \ell_{1} - \cdots - \ell_{n}}}
\,\,\,\,{\dd z \over 2\pi\ic}
\\[5mm] = &\
\oint_{\verts{z} = 1^{-}}\,\,\,{1 \over z^{s + 1}}
\,\bracks{\sum_{\ell = 0}^{\infty}{z^{\ell} \over \pars{\ell!}^{2}}}^{n}
{\dd z \over 2\pi\ic} =
\oint_{\verts{z} = 1^{-}}\,\,\,{\mrm{I}_{0}^{n}\pars{2\root{z}} \over z^{s + 1}}
{\dd z \over 2\pi\ic} =\ \bbox[15px,#ffe,border:2px dotted navy]{\ds{%
\bracks{z^{s}}\mrm{I}_{0}^{n}\pars{2\root{z}}}}
\end{align}

$\ds{\mrm{I}_{\nu}\pars{z}}$ is a
  Modified Bessel Function.

