I know that the Gamma function is related with harmonic numbers since in the sum $$1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n}$$ for $n>1$, one can get particular values of the Gamma function as the denominators of such sum (the numerators thus are the unsigned Stirling numbers of first kind $s(n+1,2):=a(n+1)$, see if you want the Sloane's sequence A000254).
Also I know that particular values of the derivative of the Gamma function are related with harmonic numbers, Wikipedia said that there is a relation between the derivative of the Gamma function and harmonic numbers $$\Gamma'(m+1)=m!\left(-\gamma+H_m\right)$$ (see Properties in the article dedicated to Gamma function).
Question. Is there relation between particular values of a derivative of high order of the Gamma function and harmonic numbers? For example can you do an specilization or the second or third derivative of the Gamma function yielding an identity with harmonic numbers (you can use generalized harmonic numbers). Thanks in advance.