  We present a compact formulation of the orthogonal Weingarten formula, with
the traditional quantity $I(i_1,...,i_{2k}:j_1,...,j_{2k}) =
\int_{O_n}u_{i_1j_1} ... u_{i_{2k}j_{2k}} du$ replaced by the more advanced
quantity $I(a)=\int_{O_n}\Pi u_{ij}^{a_{ij}} du$, depending on a matrix of
exponents $a\in M_n(\mathbb N)$. Among consequences, we establish a number of
basic facts regarding the integrals $I(a)$: vanishing conditions, possible
poles, asymptotic behavior.
