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Modified Bernoulli Number

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The modified Bernoulli numbers b_(2n) have generating function

sum_(n=0)^(infty)b_(2n)x^(2n)=1/2ln((e^(x/2)-e^(-x/2))/x)
(1)
=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).

They have closed forms

b_0=0
(3)
b_(2n)=(B_(2n))/(4n(2n)!)
(4)

for positive integers n, and b_(2n-1)=0. Here B_(2n) is a Bernoulli number and (2n)! is a factorial.

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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