Definition:Bernoulli Numbers/Sequence
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).
Given that there are a number of entries in 1983: François Le Lionnais and Jean Brette: Les Nombres Remarquables directly relating to individual entries in this table, and that there may be interesting things to say about each of the individual entries, it's worth considering whether to split each of the entries into its own page.
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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
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Entry: Bernoulli number
- Weisstein, Eric W. "Bernoulli Numbers." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/BernoulliNumber.html
