  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.
