Bernoulli Number of the Second Kind

A number defined by , where  is a Bernoulli polynomial of the second kind (Roman 1984, p. 294), also called Cauchy numbers of the first kind. The first few for , 1, 2, ... are 1, 1/2, , 1/4, , 9/4, ... (OEIS A006232 and A006233). They are given by

where  is a falling factorial, and have exponential generating function

REFERENCES:

Comtet, L. Advanced Combinatorics: The Art of Finite and Infinite Expansions, rev. enl. ed. Dordrecht, Netherlands: Reidel, p. 294, 1974.

Jeffreys, H. and Jeffreys, B. S. Methods of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge University Press, p. 259, 1988.

Roman, S. The Umbral Calculus. New York: Academic Press, p. 114, 1984.

Sloane, N. J. A. Sequences A006232/M5067 and A006233/M1558 in "The On-Line Encyclopedia of Integer Sequences."