Editing Bernoulli number (section)

==Notation==
The superscript {{math|±}} used in this article distinguishes the two sign conventions for Bernoulli numbers. Only the {{math|''n'' {{=}} 1}} term is affected: 
* {{math|''B''{{su|p=−|b=''n''}} }} with {{math|''B''{{su|p=−|b=1}} {{=}}  −{{sfrac|1|2}} }} ({{OEIS2C|id=A027641}} / {{OEIS2C|id=A027642}}) is the sign convention prescribed by [[NIST]] and most modern textbooks.{{sfnp|Arfken|1970|p=278}}
* {{math|''B''{{su|p=+|b=''n''}}}} with {{math|''B''{{su|p=+|b=1}} {{=}} +{{sfrac|1|2}} }} ({{OEIS2C|id=A164555}} / {{OEIS2C|id=A027642}}) was used in the older literature,{{r|Weisstein2016}} and (since 2022) by [[Donald Knuth]]<ref>[[Donald Knuth]] (2022), [https://www-cs-faculty.stanford.edu/~knuth/news22.html Recent News (2022): Concrete Mathematics and Bernoulli]. {{blockquote|But last year I took a close look at Peter Luschny's Bernoulli manifesto, where he gives more than a dozen good reasons why the value of $B_1$ should really be plus one-half. He explains that some mathematicians of the early 20th century had unilaterally changed the conventions, because some of their formulas came out a bit nicer when the negative value was used. It was their well-intentioned but ultimately poor choice that had led to what I'd been taught in the 1950s. […]  By now, hundreds of books that use the “minus-one-half” convention have unfortunately been written. Even worse, all the major software systems for symbolic mathematics have that 20th-century aberration deeply embedded. Yet Luschny convinced me that we have all been wrong, and that it's high time to change back to the correct definition before the situation gets even worse.
}}</ref> following Peter Luschny's "Bernoulli Manifesto".<ref>Peter Luschny (2013), [http://luschny.de/math/zeta/The-Bernoulli-Manifesto.html The Bernoulli Manifesto]</ref>

In the formulas below, one can switch from one sign convention to the other with the relation <math>B_n^{+}=(-1)^n B_n^{-}</math>, or for integer {{mvar|n}} = 2 or greater, simply ignore it.

Since {{math|''B''{{sub|''n''}} {{=}} 0}} for all odd {{math|''n'' > 1}}, and many formulas only involve even-index Bernoulli numbers, a few authors write "{{math|''B''{{sub|''n''}}}}" instead of {{math|''B''{{sub|2''n''}}&nbsp;}}. This article does not follow that notation.