Editing Bernoulli number (section)

==Related sequences==
The arithmetic mean of the first and the second Bernoulli numbers are the associate Bernoulli numbers: 
{{math|1=''B''<sub>0</sub> = 1}}, {{math|1=''B''<sub>1</sub> = 0}}, {{math|1=''B''<sub>2</sub> = {{sfrac|1|6}}}}, {{math|1=''B''<sub>3</sub> = 0}}, {{math|1=''B''<sub>4</sub> = −{{sfrac|1|30}}}}, {{OEIS2C|id=A176327}} / {{OEIS2C|id=A027642}}. Via the second row of its inverse Akiyama–Tanigawa transform {{OEIS2C|id=A177427}}, they lead to Balmer series {{OEIS2C|id=A061037}} / {{OEIS2C|id=A061038}}.

The Akiyama–Tanigawa algorithm applied to {{OEIS2C|id=A060819}} ({{math|''n'' + 4}}) / {{OEIS2C|id=A145979}} ({{mvar|n}}) leads to the Bernoulli numbers {{OEIS2C|id=A027641}} / {{OEIS2C|id=A027642}}, {{OEIS2C|id=A164555}} / {{OEIS2C|id=A027642}}, or {{OEIS2C|id=A176327}} {{OEIS2C|id=A176289}} without {{math|''B''<sub>1</sub>}}, named intrinsic Bernoulli numbers {{math|''B''<sub>''i''</sub>(''n'')}}.

:{| style="text-align:center; padding-left; padding-right: 2em;"
|-
|1||{{sfrac|5|6}}||{{sfrac|3|4}}||{{sfrac|7|10}}||{{sfrac|2|3}}
|-
|{{sfrac|1|6}}||{{sfrac|1|6}}||{{sfrac|3|20}}||{{sfrac|2|15}}||{{sfrac|5|42}}
|-
|0||{{sfrac|1|30}}||{{sfrac|1|20}}||{{sfrac|2|35}}||{{sfrac|5|84}}
|-
|−{{sfrac|1|30}}||−{{sfrac|1|30}}||−{{sfrac|3|140}}||−{{sfrac|1|105}}||0
|-
|0||−{{sfrac|1|42}}||−{{sfrac|1|28}}||−{{sfrac|4|105}}||−{{sfrac|1|28}}
|}

Hence another link between the intrinsic Bernoulli numbers and the Balmer series via {{OEIS2C|id=A145979}} ({{math|''n''}}).

{{OEIS2C|id=A145979}} ({{math|''n'' − 2}}) = 0, 2, 1, 6,... is a permutation of the non-negative numbers.

The terms of the first row are f(n) = {{math|{{sfrac|1|2}} + {{sfrac|1|''n'' + 2}}}}. 2, f(n) is an autosequence of the second kind. 3/2, f(n) leads by its inverse binomial transform to 3/2 −1/2 1/3 −1/4 1/5 ... = 1/2 + log 2.

Consider g(n) = 1/2 – 1 / (n+2) = 0, 1/6, 1/4, 3/10, 1/3. The Akiyama-Tanagiwa transforms gives:

:{| style="text-align:center; padding-left; padding-right:2em;"
|-
|0||{{sfrac|1|6}}||{{sfrac|1|4}}||{{sfrac|3|10}}||{{sfrac|1|3}}||{{sfrac|5|14}}||...
|-
|−{{sfrac|1|6}}||−{{sfrac|1|6}}||−{{sfrac|3|20}}||−{{sfrac|2|15}}||−{{sfrac|5|42}}||−{{sfrac|3|28}}||...
|-
|0||−{{sfrac|1|30}}||−{{sfrac|1|20}}||−{{sfrac|2|35}}||−{{sfrac|5|84}}||−{{sfrac|5|84}}||...
|-
|{{sfrac|1|30}}||{{sfrac|1|30}}||{{sfrac|3|140}}||{{sfrac|1|105}}||0||−{{sfrac|1|140}}||...
|}

0, g(n), is an autosequence of the second kind.

Euler {{OEIS2C|id=A198631}} ({{math|''n''}}) / {{OEIS2C|id=A006519}} ({{math|''n'' + 1}}) without the second term ({{sfrac|1|2}}) are the fractional intrinsic Euler numbers {{math|''E''<sub>''i''</sub>(''n'') {{=}} 1, 0, −{{sfrac|1|4}}, 0, {{sfrac|1|2}}, 0, −{{sfrac|17|8}}, 0, ...}} The corresponding Akiyama transform is:

:{| style="text-align:center; padding-left; padding-right: 2em;"
|-
|1||1||{{sfrac|7|8}}||{{sfrac|3|4}}||{{sfrac|21|32}}
|-
|0||{{sfrac|1|4}}||{{sfrac|3|8}}||{{sfrac|3|8}}||{{sfrac|5|16}}
|-
|−{{sfrac|1|4}}||−{{sfrac|1|4}}||0||{{sfrac|1|4}}||{{sfrac|25|64}}
|-
|0||−{{sfrac|1|2}}||−{{sfrac|3|4}}||−{{sfrac|9|16}}||−{{sfrac|5|32}}
|-
|{{sfrac|1|2}}||{{sfrac|1|2}}||−{{sfrac|9|16}}||−{{sfrac|13|8}}||−{{sfrac|125|64}}
|}

The first line is {{math|''Eu''(''n'')}}. {{math|''Eu''(''n'')}} preceded by a zero is an autosequence of the first kind. It is linked to the Oresme numbers. The numerators of the second line are {{OEIS2C|id=A069834}} preceded by 0. The difference table is:

:{| style="text-align:center; padding-left; padding-right: 2em;"
|-
|0||1||1||{{sfrac|7|8}}||{{sfrac|3|4}}||{{sfrac|21|32}}||{{sfrac|19|32}}
|-
|1||0||−{{sfrac|1|8}}||−{{sfrac|1|8}}||−{{sfrac|3|32}}||−{{sfrac|1|16}}||−{{sfrac|5|128}}
|-
|−1||−{{sfrac|1|8}}||0||{{sfrac|1|32}}||{{sfrac|1|32}}||{{sfrac|3|128}}||{{sfrac|1|64}}
|}