Definition:Bernoulli Numbers/Sequence

From ProofWiki
Jump to navigation Jump to search

Definition

The sequence of Bernoulli numbers begins:

\(\displaystyle B_0\) \(=\) \(\displaystyle 1\)
\(\displaystyle B_1\) \(=\) \(\, \displaystyle - \, \) \(\displaystyle \dfrac 1 2\)
\(\displaystyle B_2\) \(=\) \(\displaystyle \dfrac 1 6\)
\(\displaystyle B_4\) \(=\) \(\, \displaystyle - \, \) \(\displaystyle \dfrac 1 {30}\)
\(\displaystyle B_6\) \(=\) \(\displaystyle \dfrac 1 {42}\)
\(\displaystyle B_8\) \(=\) \(\, \displaystyle - \, \) \(\displaystyle \dfrac 1 {30}\)
\(\displaystyle B_{10}\) \(=\) \(\displaystyle \dfrac 5 {66}\)
\(\displaystyle B_{12}\) \(=\) \(\, \displaystyle - \, \) \(\displaystyle \dfrac {691} {2730}\)

The odd index Bernoulli numbers, apart from $B_1$, are all equal to $0$.

The numerators form sequence A027641 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

The denominators form sequence A027642 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).



Sources