  Dielectronic recombination (DR) is the dominant mode of recombination in
magnetically confined fusion plasmas for intermediate to low-charged ions of W.
Complete, final-state resolved partial isonuclear W DR rate coefficient data is
required for detailed collisional-radiative modelling for such plasmas in
preparation for the upcoming fusion experiment ITER. To realize this
requirement, we continue {\it The Tungsten Project} by presenting our
calculations for tungsten ions W$^{55+}$ to W$^{38+}$. As per our prior
calculations for W$^{73+}$ to W$^{56+}$, we use the collision package {\sc
autostructure} to calculate partial and total DR rate coefficients for all
relevant core-excitations in intermediate coupling (IC) and configuration
average (CA) using $\kappa$-averaged relativistic wavefunctions. Radiative
recombination (RR) rate coefficients are also calculated for the purpose of
evaluating ionization fractions. Comparison of our DR rate coefficients for
W$^{46+}$ with other authors yields agreement to within 7-19\% at peak
abundance verifying the reliability of our method. Comparison of partial DR
rate coefficients calculated in IC and CA yield differences of a factor
$\sim{2}$ at peak abundance temperature, highlighting the importance of
relativistic configuration mixing. Large differences are observed between
ionization fractions calculated using our recombination rate coefficient data
and that of P\"{u}tterich~\etal [Plasma Phys. and Control. Fusion 50 085016,
(2008)]. These differences are attributed to deficiencies in the average-atom
method used by the former to calculate their data.
