In the book of L. Comtet
[Advanced combinatorics. The art of finite and infinite expansions. Dordrecht / Boston, D. Reidel Publishing Company. (1974; Zbl 0283.05001
)], two kinds of Cauchy numbers are introduced. The present author studies sums of products of these numbers, which have not been studied earlier, by the method of coefficients. Among other results, he derives identities involving these sums and various numbers: Stirling, generalized Bernoulli, generalized Euler, Lah, and harmonic.