The orthogonal Weingarten formula in compact form

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 $I(a)={\int }_{O_n}\mathrm{\Pi }{u}_{ij}^{a_{ij}}du$, 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.