Powers and inverses of special upper triangular matrices

Motivated by old and recent works on matrix powers and their applications on combinatorial sequences, for given $\left(a_{n};n\geq 1\right) $ sequence of complex numbers and power series $\varphi,$ $g,$ $h$ such that $a_{n}\neq 0,$ $n\geq 1,$ $\varphi \left( 0\right) =0$ and $\varphi ^{\prime }\left( 0\right) g\left( 0\right) h\left( 0\right) \neq 0,$ we give in this paper the $s$-th power and the inverse of the upper triangular matrices having $\left( k,n\right) $-th entry in the form \begin{equation*} \frac{a_{k}}{a_{n}}\frac{1}{k!}\left( \frac{d}{dt}\right)_{\! \!t=0}^{\! \!n-1}\left[ \left( h\left( t\right) \right) ^{n}\frac{d}{dt}\left( \left( \varphi \left( t\right) \right) ^{k}g\left( t\right) \right) \right]. \end{equation*}