Definition:Bernoulli Numbers/Sequence

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Definition

The sequence of Bernoulli numbers begins:

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

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