QUESTION:
I need to know how to compute the partial sum $$\sum_{k=1}^n \frac{H_{k+1}^2-H_{k+1}^{(2)}}{k+2}$$ in terms of the generalized harmonic numbers $H_n^{(m)}$.
CONTEXT:
This problem arose because I was trying to compute the coefficients of the Taylor series of the function $$f(z)=\ln^4(1-z)$$ And I have already computed that $$\sum_{k=1}^\infty \frac{H_{k+1}^2-H_{k+1}^{(2)}}{k+2}z^{k+2}=\frac{\ln^3(1-z)}{3}$$