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We show that if a group contains $\mathbb{Z}^n \times F_m$ as a finite-indexsubgroup, then its cogrowth series is the diagonal of a rational function forevery generating set. This answers a question of Pak and Soukup on the cogrowthof virtually abelian groups; and generalises a result by Elder, Rechnitzer,Janse van Rensburg, and Wong on the cogrowth series of the Baumslag-Solitargroups $\mathrm{BS}(N,N)$.

On groups whose cogrowth series is the diagonal of a rational series