  We prove Rogers-Ramanujan type identities for the Nahm sums associated with
the tadpole Cartan matrix of rank $3$. These identities reveal the modularity
of these sums, and thereby we confirm a conjecture of Penn, Calinescu and the
first author in this case. We show that these Nahm sums together with some
shifted sums can be combined into a vector-valued modular function on the full
modular group. We also present some conjectures for a general rank.
