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In the paper, by establishing a new and explicit formula for computing the$n$-th derivative of the reciprocal of the logarithmic function, the authorpresents new and explicit formulas for calculating Bernoulli numbers of thesecond kind and Stirling numbers of the first kind. As consequences of theseformulas, a recursion for Stirling numbers of the first kind and a newrepresentation of the reciprocal of the factorial $n!$ are derived. Finally,the author finds several identities and integral representations relating toStirling numbers of the first kind.

Explicit formulas for computing Bernoulli numbers of the second kind and Stirling numbers of the first kind