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
- 1983: François Le Lionnais and Jean Brette: Les Nombres Remarquables ... (previous) ... (next): $0,0757575757575 \ldots$
- 1983: François Le Lionnais and Jean Brette: Les Nombres Remarquables ... (previous) ... (next): $0,1666666666666 \ldots$
- 1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {B}.20$: The Bernoulli Numbers and some Wonderful Discoveries of Euler
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Entry: Bernoulli numbers
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Entry: Bernoulli numbers
- Weisstein, Eric W. "Bernoulli Numbers." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/BernoulliNumber.html