Definition:Bernoulli Numbers/Archaic Form/Sequence

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Definition

The sequence of old style Bernoulli numbers begins:

\(\ds B_1^*\) \(=\) \(\ds \dfrac 1 6\) \(\ds = B_2\)
\(\ds B_2^*\) \(=\) \(\ds \dfrac 1 {30}\) \(\ds = -B_4\)
\(\ds B_3^*\) \(=\) \(\ds \dfrac 1 {42}\) \(\ds = B_6\)
\(\ds B_4^*\) \(=\) \(\ds \dfrac 1 {30}\) \(\ds = -B_8\)
\(\ds B_5^*\) \(=\) \(\ds \dfrac 5 {66}\) \(\ds = B_{10}\)
\(\ds B_6^*\) \(=\) \(\ds \dfrac {691} {2730}\) \(\ds = -B_{12}\)
\(\ds B_7^*\) \(=\) \(\ds \dfrac 7 6\) \(\ds = B_{14}\)
\(\ds B_8^*\) \(=\) \(\ds \dfrac {3617} {510}\) \(\ds = -B_{16}\)
\(\ds B_9^*\) \(=\) \(\ds \dfrac {43 \, 867} {798}\) \(\ds = B_{18}\)
\(\ds B_{10}^*\) \(=\) \(\ds \dfrac {174 \, 611} {330}\) \(\ds = -B_{20}\)
\(\ds B_{11}^*\) \(=\) \(\ds \dfrac {854 \, 513} {138}\) \(\ds = B_{22}\)
\(\ds B_{12}^*\) \(=\) \(\ds \dfrac {236 \, 364 \, 091} {2730}\) \(\ds = -B_{24}\)

where $B_2, B_4, \ldots$ are the standard form Bernoulli numbers.

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


Sources