Definition:Bernoulli Numbers/Archaic Form/Sequence

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Definition

The sequence of old style Bernoulli numbers begins:

\(\displaystyle B_1^*\) \(=\) \(\displaystyle \dfrac 1 6\)
\(\displaystyle B_2^*\) \(=\) \(\displaystyle \dfrac 1 {30}\)
\(\displaystyle B_3^*\) \(=\) \(\displaystyle \dfrac 1 {42}\)
\(\displaystyle B_4^*\) \(=\) \(\displaystyle \dfrac 1 {30}\)
\(\displaystyle B_5^*\) \(=\) \(\displaystyle \dfrac 5 {66}\)
\(\displaystyle B_6^*\) \(=\) \(\displaystyle \dfrac {691} {2730}\)
\(\displaystyle B_7^*\) \(=\) \(\displaystyle \dfrac 7 6\)
\(\displaystyle B_8^*\) \(=\) \(\displaystyle \dfrac {3617} {510}\)
\(\displaystyle B_9^*\) \(=\) \(\displaystyle \dfrac {43 \, 867} {798}\)
\(\displaystyle B_{10}^*\) \(=\) \(\displaystyle \dfrac {174 \, 611} {330}\)
\(\displaystyle B_{11}^*\) \(=\) \(\displaystyle \dfrac {854 \, 513} {138}\)
\(\displaystyle B_{12}^*\) \(=\) \(\displaystyle \dfrac {236 \, 364 \, 091} {2730}\)

The denominators form sequence A002445 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


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