\alpha=f_{1}^{*}(A)=a-2b+6c-24d+120e-720f
\\
\\ \beta=f_{2}^{*}(A)=b-3c+12d-60e+360f
\\
\\ \gamma=f_{3}^{*}(A)=c-4d+20e-120f
\\
\\ \delta=f_{4}^{*}(A)=d-5e+30f
\\
\\ \epsilon=f_{5}^{*}(A)=e-6f
\\
\\ \omega=f_{6}(A)=f