Log-convexity and the cycle index polynomials with relation to compound Poisson distributions

We extend the exponential formula by Bender and Canfield (1996), which relates log-concavity and the cycle index polynomials. The extension clarifies the log-convexity relation. The proof is by noticing the property of a compound Poisson distribution together with its moment generating function. We also give a combinatorial proof of extended "log-convex part" referring Bender and Canfield's approach, where the formula by Bruijn and Erd\"os (1953) is additionally exploited. The combinatorial approach yields richer structural results more than log-convexity. Furthermore, we consider normal and binomial convolutions of sequences which satisfy the exponential formula. The operations generate interesting examples which are not included in well known laws about log-concavity/convexity.