Sum over partitions

Question

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.

Answer 1

$\newcommand{\bbx}[1]{\,\bbox[8px,border:1px groove navy]{{#1}}\,} \newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace} \newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack} \newcommand{\dd}{\mathrm{d}} \newcommand{\ds}[1]{\displaystyle{#1}} \newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,} \newcommand{\ic}{\mathrm{i}} \newcommand{\mc}[1]{\mathcal{#1}} \newcommand{\mrm}[1]{\mathrm{#1}} \newcommand{\pars}[1]{\left(\,{#1}\,\right)} \newcommand{\partiald}[3][]{\frac{\partial^{#1} #2}{\partial #3^{#1}}} \newcommand{\root}[2][]{\,\sqrt[#1]{\,{#2}\,}\,} \newcommand{\totald}[3][]{\frac{\mathrm{d}^{#1} #2}{\mathrm{d} #3^{#1}}} \newcommand{\verts}[1]{\left\vert\,{#1}\,\right\vert}$ \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.