An integrality phenomenon

We prove a general statement about the integrality of the sequences generated by a recursion of the following form: $n{u}_n$ equals a linear combination of $u_{n-1},u_{n-2},\dots ,u_0$ with polynomial coefficients in $n$ of special form. This includes a conjectural integrality of the sequence related to the Hörmander-Bernhardsson extremal function, for which we further give a direct proof as well.