  This paper is devoted to the family $\{G_n\}$ of hypergeometric series of any
finite number of variables, the coefficients being the square of the
multinomial coefficients $(\ell_1+...+\ell_n)!/(\ell_1!...\ell_n!)$, where
$n\in\ZZ_{\ge 1}$. All these series belong to the family of the general
Appell-Lauricella's series. It is shown that each function $G_n$ can be
expressed by an integral involving the previous one, $G_{n-1}$. Thus this
family can be represented by a multidimensional Euler type integral, what
suggests some explicit link with the Gelfand-Kapranov-Zelevinsky's theory of
$A$-hypergeometric systems or with the Aomoto's theory of hypermeotric
functions. The quasi-invariance of each function $G_n$ with regard to the
action of a finite number of involutions of $\CC^{*n}$ is also established.
Finally, a particular attention is reserved to the study of the functions $G_2$
and $G_3$, each of which is proved to be algebraic or to be expressed by the
Legendre's elliptic function of the first kind.
