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We present some properties of the expansion coefficients $a_{mk}$ and$c_{mk}$ of a pair of dual bases,  \[ n^m = \sum_{k=2}^m c_{mk} \psi_{k}(n), \] and  \[ \psi_m(n) = n + (m-1)(n-1) B_{n-1,m-1}, \] we introduced earlier inarXiv:2207.01935v1. Here, $B_{a,b} = (a+b)!/(a!\,b!)$ is a binomialcoefficient. We extend the knowledge on the $c_{mk}$ coefficients by giving anexplicit expression for them in terms of the Stirling numbers of the secondkind. From the interchangeability of the indices of the binomial coefficient,follows the central identity we use here:  \[ \psi_m(n) - n = \psi_n(m) - m. \] With this equation, we evaluate sums ofthe form  \[ T^\alpha_m = \sum_{k=2}^m c_{mk} k^\alpha.\] Explicitly, the case $T_m^1$is handled. Furthermore, we indicate connections of $T_m^2$ and $T_m^3$ to theMersenne numbers (general integer exponent) and the OEIS entry A024023. Weconclude with a small remark on how we can represent Pythagoras' equation interms of the $a_{mk}$ coefficients.

Further Identities for $c_{mk}$- and $a_{mk}$-Weighted Sums and a Remark on a Representation of Pythagoras' Equation