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
See also Bernoulli Number ,
Bernoulli
Polynomial of the Second Kind
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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." Referenced on Wolfram|Alpha Bernoulli Number of the
Second Kind
Cite this as:
Weisstein, Eric W. "Bernoulli Number of the Second Kind." From MathWorld --A Wolfram Web Resource.
https://mathworld.wolfram.com/BernoulliNumberoftheSecondKind.html
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