Editing Bernoulli number (section)

=== Connection with Stirling numbers of the first kind ===
The two main formulas relating the unsigned [[Stirling numbers of the first kind]] {{math|<big><big>[</big></big>{{su|p=''n''|b=''m''|a=c}}<big><big>]</big></big>}} to the Bernoulli numbers (with {{math|''B''<sub>1</sub> {{=}} +{{sfrac|1|2}}}}) are

: <math> \frac{1}{m!}\sum_{k=0}^m (-1)^{k} \left[{m+1\atop k+1}\right] B_k = \frac{1}{m+1}, </math>

and the inversion of this sum (for {{math|''n'' ≥ 0}}, {{math|''m'' ≥ 0}})

: <math> \frac{1}{m!}\sum_{k=0}^m (-1)^k \left[{m+1\atop k+1}\right] B_{n+k} = A_{n,m}. </math>

Here the number {{math|''A''<sub>''n'',''m''</sub>}} are the rational Akiyama–Tanigawa numbers, the first few of which are displayed in the following table.

:{| class="wikitable" style="text-align:center"
|+ Akiyama–Tanigawa number
! {{diagonal split header|{{mvar|n}}|{{mvar|m}}}}!!0!!1!!2!!3!!4
|-
! 0 
| 1 || {{sfrac|1|2}} || {{sfrac|1|3}} || {{sfrac|1|4}} || {{sfrac|1|5}}
|-
! 1 
| {{sfrac|1|2}} || {{sfrac|1|3}} || {{sfrac|1|4}} || {{sfrac|1|5}} || ...
|-
! 2 
| {{sfrac|1|6}} || {{sfrac|1|6}} || {{sfrac|3|20}} || ... || ...
|-
! 3 
| 0 || {{sfrac|1|30}} || ... || ... || ...
|-
! 4 
| −{{sfrac|1|30}} || ... || ... || ... || ...
|}

The Akiyama–Tanigawa numbers satisfy a simple recurrence relation which can be exploited to iteratively compute the Bernoulli numbers. This leads to the algorithm shown in the section 'algorithmic description' above. See {{OEIS2C|id=A051714}}/{{OEIS2C|id=A051715}}.

An ''autosequence'' is a sequence which has its inverse binomial transform equal to the signed sequence. If the main diagonal is zeroes = {{OEIS2C|id=A000004}}, the autosequence is of the first kind. Example: {{OEIS2C|id=A000045}}, the Fibonacci numbers. If the main diagonal is the first upper diagonal multiplied by 2, it is of the second kind. Example: {{OEIS2C|id=A164555}}/{{OEIS2C|id=A027642}}, the second Bernoulli numbers (see {{OEIS2C|id=A190339}}). The Akiyama–Tanigawa transform applied to {{math|''2''<sup>−''n''</sup>}} = 1/{{OEIS2C|id=A000079}} leads to {{OEIS2C|id=A198631}} (''n'') / {{OEIS2C|id=A06519}} (''n'' + 1). Hence:

:{| class="wikitable" style="text-align:center"
|+ Akiyama–Tanigawa transform for the second Euler numbers
|-
! {{diagonal split header|{{mvar|n}}|{{mvar|m}}}} !! 0 !! 1 !! 2 !! 3 !! 4
|-
! 0
| 1 || {{sfrac|1|2}} || {{sfrac|1|4}} || {{sfrac|1|8}} || {{sfrac|1|16}}
|-
! 1 
| {{sfrac|1|2}} || {{sfrac|1|2}} || {{sfrac|3|8}} || {{sfrac|1|4}} || ...
|-
! 2 
| 0 || {{sfrac|1|4}} || {{sfrac|3|8}} || ... || ...
|-
! 3 
| −{{sfrac|1|4}} || −{{sfrac|1|4}} || ... || ... || ...
|-
! 4 
| 0 || ... || ... || ... || ...
|}

See {{OEIS2C|id=A209308}} and {{OEIS2C|id=A227577}}. {{OEIS2C|id=A198631}} ({{math|''n''}}) / {{OEIS2C|id=A006519}} ({{math|''n'' + 1}}) are the second (fractional) Euler numbers and an autosequence of the second kind.

:({{sfrac|{{OEIS2C|id=A164555}} ({{math|''n'' + 2}})|{{OEIS2C|id=A027642}} ({{math|''n'' + 2}})}} = {{math|{{sfrac|1|6}}, 0, −{{sfrac|1|30}}, 0, {{sfrac|1|42}}, ...}}) × ( {{math|{{sfrac|2<sup>''n'' + 3</sup> − 2|''n'' + 2}}}} = {{math|3, {{sfrac|14|3}}, {{sfrac|15|2}}, {{sfrac|62|5}}, 21, ...}}) = {{sfrac|{{OEIS2C|id=A198631}} ({{math|''n'' + 1}})|{{OEIS2C|id=A006519}} ({{math|''n'' + 2}})}} = {{math|{{sfrac|1|2}}, 0, −{{sfrac|1|4}}, 0, {{sfrac|1|2}}, ...}}.

Also valuable for {{OEIS2C|id=A027641}} / {{OEIS2C|id=A027642}} (see [[#Connection with Worpitzky numbers|Connection with Worpitzky numbers]]).