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Modified Bernoulli Number

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A number b_(2n) having generating function

sum_(n=0)^(infty)b_(2n)x^(2n)=1/2ln((e^(x/2)-e^(-x/2))/(1/2x))
(1)
=1/2ln2+1/(48)x^2-1/(5760)x^4+1/(362880)x^6-....
(2)

For n=1, 2, ..., the denominators are 48, 5760, 362880, 19353600, ... (OEIS A057868).

It has closed form

b_0=1/2ln2
(3)
b_n=(B_n)/(2n^2Gamma(n))
(4)

and b_(2k-1)=0, where B_n is a Bernoulli number and Gamma(n) a gamma function.

See also

Bernoulli Number, Kontsevich Integral

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References

Sloane, N. J. A. Sequence A057868 in "The On-Line Encyclopedia of Integer Sequences."

Referenced on Wolfram|Alpha

Modified Bernoulli Number

Cite this as:

Weisstein, Eric W. "Modified Bernoulli Number." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ModifiedBernoulliNumber.html

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