The weighted mean matrix with weight sequence $w_n=2n+1$ is a hyponormal operator on ${\mathrm{\ell }}^2$

A weighted mean matrix whose weight sequence is linear with positive coefficients is shown to be a posinormal operator on ${\mathrm{\ell }}^2$. This operator is also shown to be coposinormal, so it and its adjoint have the same null space and the same range. The posinormality result leads to a proof that the weighted mean matrix associated with the sequence of odd positive integers is hyponormal, as well as a conjecture regarding a more general linear case.