We expand on the remark by Andrews on the importance of infinite sums and products in combinatorics. Let \(\{g_d(n)\}_{d\geq 0,n \geq 1}\) be the double sequences \(\sigma_d(n)= \sum_{\ell \mid n} \ell^d\) or \(\psi_d(n)= n^d\). We associate double sequences \(\left\{ p^{g_{d} }\left( n\right) \right\}\) and \(\left\{ q^{g_{d} }\left( n\right) \right\} \), defined as the coefficients of \begin{eqnarray*} \sum_{n=0}^{\infty} p^{g_{d} }\left( n\right) \, t^{n} & := & \prod_{n=1}^{\infty} \left( 1 - t^{n} \right)^{-\frac{ \sum_{\ell \mid n} \mu(\ell) \, g_d(n/\ell) }{n} }, \\ \sum_{n=0}^{\infty} q^{g_{d} }\left( n\right) \, t^{n} & := & \frac{1}{1 - \sum_{n=1}^{\infty} g_d(n) \, t^{n} }. \end{eqnarray*} These coefficients are related to the number of partitions \(\mathrm{p}\left( n\right) = p^{\sigma _{1 }}\left ( n\right) \), plane partitions \(pp\left( n\right) = p^{\sigma _{2 }}\left( n\right) \) of \(n\), and Fibonacci numbers \(F_{2n} = q^{\psi _{1 }}\left( n\right) \). Let \(n \geq 3\) and let \(n \equiv 0 \pmod{3}\). Then the coefficients are log-concave at \(n\) for almost all \(d\) in the exponential and geometric cases. The coefficients are not log-concave for almost all \(d\) in both cases, if \(n \equiv 2 \pmod{3}\). Let \(n\equiv 1 \pmod{3}\). Then the log-concave property flips for almost all \(d\).