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
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 21$: Table of First Few Bernoulli and Euler Numbers
- Weisstein, Eric W. "Bernoulli Number." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/BernoulliNumber.html