The generating function of the sequence $W_n(z)$ (shifted) is $$w(t,z):=(1-2zt+t^2)^{-\frac{1}{2}}\log\bigg(\frac{-z+t+(1-2zt+t^2)^\frac{1}{2}}{1-z}\bigg) =\sum_{n=1}^\infty\\ W_{n-1} (z)\\ t^n\\ .$$ It verifies a simple linear first order differential equation: $$(1-2zt+t^2)w_t + (t-z)w=1$$ that translates into a three-terms linear recurrence for the $W_n\\ $:
$$(n+1)W_n=(2n+1)zW_{n-1}-nW_{n-2}\qquad (n\ge2)$$
with the initial contitions $W_0=1$ and $W_1:=\frac{3}{2}z\\ .$