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A sequence of rational numbers as a generalization of the sequence ofBernoulli numbers is introduced. Sums of products involving the terms of thisgeneralized sequence are then obtained using an application of the Fa\`a diBruno's formula. These sums of products are analogous to the higher orderBernoulli numbers and are used to develop the closed form expressions for thesums of products involving the power sums $\displaystyle\Psi_k(x,n):=\sum_{d|n}\mu(d)d^k S_k(\frac{x}{d}), n\in\mathbb{Z}^+$ which aredefined via the M\"obius function $\mu$ and the usual power sum $S_k(x)$ of areal or complex variable $x.$ The power sum $S_k(x)$ is expressible in terms ofthe well known Bernoulli polynomials by $\displaystyleS_k(x):=\frac{B_{k+1}(x+1)-B_{k+1}(0)}{k+1}.$

Sums of products involving power sums of $\varphi(n)$ integers