Editing Bernoulli number (section)

== Connections with combinatorial numbers ==
The connection of the Bernoulli number to various kinds of combinatorial numbers is based on the classical theory of finite differences and on the combinatorial interpretation of the Bernoulli numbers as an instance of a fundamental combinatorial principle, the [[inclusion–exclusion principle]].

=== Connection with Worpitzky numbers ===
The definition to proceed with was developed by Julius Worpitzky in 1883. Besides elementary arithmetic only the factorial function {{math|''n''!}} and the power function {{math|''k<sup>m</sup>''}} is employed. The signless Worpitzky numbers are defined as

: <math> W_{n,k}=\sum_{v=0}^k (-1)^{v+k} (v+1)^n \frac{k!}{v!(k-v)!} . </math>

They can also be expressed through the [[Stirling numbers of the second kind]]

: <math> W_{n,k}=k! \left\{ {n+1\atop k+1} \right\}.</math>

A Bernoulli number is then introduced as an inclusion–exclusion sum of Worpitzky numbers weighted by the [[Harmonic progression (mathematics)|harmonic sequence]] 1,&nbsp;{{sfrac|1|2}},&nbsp;{{sfrac|1|3}},&nbsp;...

: <math> B_{n}=\sum_{k=0}^n (-1)^k \frac{W_{n,k}}{k+1}\ =\ \sum_{k=0}^n \frac{1}{k+1} \sum_{v=0}^k (-1)^v (v+1)^n {k \choose v}\ . </math>

:{{math|1=''B''<sub>0</sub> = 1}}
:{{math|1=''B''<sub>1</sub> = 1 − {{sfrac|1|2}}}}
:{{math|1=''B''<sub>2</sub> = 1 − {{sfrac|3|2}} + {{sfrac|2|3}}}}
:{{math|1=''B''<sub>3</sub> = 1 − {{sfrac|7|2}} + {{sfrac|12|3}} − {{sfrac|6|4}}}}
:{{math|1=''B''<sub>4</sub> = 1 − {{sfrac|15|2}} + {{sfrac|50|3}} − {{sfrac|60|4}} + {{sfrac|24|5}}}}
:{{math|1=''B''<sub>5</sub> = 1 − {{sfrac|31|2}} + {{sfrac|180|3}} − {{sfrac|390|4}} + {{sfrac|360|5}} − {{sfrac|120|6}}}}
:{{math|1=''B''<sub>6</sub> = 1 − {{sfrac|63|2}} + {{sfrac|602|3}} − {{sfrac|2100|4}} + {{sfrac|3360|5}} − {{sfrac|2520|6}} + {{sfrac|720|7}}}}

This representation has {{math|''B''{{su|p=+|b=1}} {{=}} +{{sfrac|1|2}}}}.

Consider the sequence {{math|''s<sub>n</sub>''}}, {{math|''n'' ≥ 0}}. From Worpitzky's numbers {{OEIS2C|id=A028246}}, {{OEIS2C|id=A163626}} applied to {{math|''s''<sub>0</sub>, ''s''<sub>0</sub>, ''s''<sub>1</sub>, ''s''<sub>0</sub>, ''s''<sub>1</sub>, ''s''<sub>2</sub>, ''s''<sub>0</sub>, ''s''<sub>1</sub>, ''s''<sub>2</sub>, ''s''<sub>3</sub>, ...}} is identical to the Akiyama–Tanigawa transform applied to {{math|''s<sub>n</sub>''}} (see [[#Connection with Stirling numbers of the first kind|Connection with Stirling numbers of the first kind]]). This can be seen via the table:

:{| style="text-align:center"
|+ '''Identity of<br/>Worpitzky's representation and Akiyama–Tanigawa transform'''
|-
|1|| || || || || ||0||1|| || || || ||0||0||1|| || || ||0||0||0||1|| || ||0||0||0||0||1|| 
|-
|1||−1|| || || || ||0||2||−2|| || || ||0||0||3||−3|| || ||0||0||0||4||−4|| || || || || || || 
|-
|1||−3||2|| || || ||0||4||−10||6|| || ||0||0||9||−21||12|| || || || || || || || || || || || || 
|-
|1||−7||12||−6|| || ||0||8||−38||54||−24|| || || || || || || || || || || || || || || || || || || 
|-
|1||−15||50||−60||24|| || || || || || || || || || || || || || || || || || || || || || || || || 
|-
|}

The first row represents {{math|''s''<sub>0</sub>, ''s''<sub>1</sub>, ''s''<sub>2</sub>, ''s''<sub>3</sub>, ''s''<sub>4</sub>}}.

Hence for the second fractional Euler numbers {{OEIS2C|id=A198631}} ({{math|''n''}}) / {{OEIS2C|id=A006519}} ({{math|''n'' + 1}}):

:{{math|1= ''E''<sub>0</sub> = 1}}
:{{math|1= ''E''<sub>1</sub> = 1 − {{sfrac|1|2}}}}
:{{math|1= ''E''<sub>2</sub> = 1 − {{sfrac|3|2}} + {{sfrac|2|4}}}}
:{{math|1= ''E''<sub>3</sub> = 1 − {{sfrac|7|2}} + {{sfrac|12|4}} − {{sfrac|6|8}}}}
:{{math|1= ''E''<sub>4</sub> = 1 − {{sfrac|15|2}} + {{sfrac|50|4}} − {{sfrac|60|8}} + {{sfrac|24|16}}}}
:{{math|1= ''E''<sub>5</sub> = 1 − {{sfrac|31|2}} + {{sfrac|180|4}} − {{sfrac|390|8}} + {{sfrac|360|16}} − {{sfrac|120|32}}}}
:{{math|1= ''E''<sub>6</sub> = 1 − {{sfrac|63|2}} + {{sfrac|602|4}} − {{sfrac|2100|8}} + {{sfrac|3360|16}} − {{sfrac|2520|32}} + {{sfrac|720|64}}}}

A second formula representing the Bernoulli numbers by the Worpitzky numbers is for {{math|''n'' ≥ 1}}

: <math> B_n=\frac n {2^{n+1}-2}\sum_{k=0}^{n-1} (-2)^{-k}\, W_{n-1,k} . </math>

The simplified second Worpitzky's representation of the second Bernoulli numbers is:

{{OEIS2C|id=A164555}} ({{math|''n'' + 1}}) / {{OEIS2C|id=A027642}}({{math|''n'' + 1}}) = {{math|{{sfrac|''n'' + 1|2<sup>''n'' + 2</sup> − 2}}}} × {{OEIS2C|id=A198631}}({{math|''n''}}) / {{OEIS2C|id=A006519}}({{math|''n'' + 1}})

which links the second Bernoulli numbers to the second fractional Euler numbers. The beginning is:

:{{math|{{sfrac|1|2}}, {{sfrac|1|6}}, 0, −{{sfrac|1|30}}, 0, {{sfrac|1|42}}, ... {{=}} ({{sfrac|1|2}}, {{sfrac|1|3}}, {{sfrac|3|14}}, {{sfrac|2|15}}, {{sfrac|5|62}}, {{sfrac|1|21}}, ...) × (1, {{sfrac|1|2}}, 0, −{{sfrac|1|4}}, 0, {{sfrac|1|2}}, ...)}}

The numerators of the first parentheses are {{OEIS2C|id=A111701}} (see [[#Connection with Stirling numbers of the first kind|Connection with Stirling numbers of the first kind]]).

=== Connection with Stirling numbers of the second kind ===

If one defines the [[Bernoulli polynomials]] {{math|''B<sub>k</sub>''(''j'')}} as:{{r|Rademacher1973}}

:<math> B_k(j)=k\sum_{m=0}^{k-1}\binom{j}{m+1}S(k-1,m)m!+B_k </math>

where {{math|''B<sub>k</sub>''}} for {{math|''k'' {{=}} 0, 1, 2,...}} are the Bernoulli numbers,
and {{math|''S(k,m)''}} is a [[Stirling numbers of the second kind|Stirling number of the second kind]].

One also has the following for Bernoulli polynomials,{{r|Rademacher1973}}

:<math>  B_k(j)=\sum_{n=0}^k \binom{k}{n} B_n j^{k-n}. </math>

The coefficient of {{mvar|j}} in {{math|<big><big>(</big></big>{{su|p=''j''|b=''m'' + 1|a=c}}<big><big>)</big></big>}} is {{math|{{sfrac|(−1)<sup>''m''</sup>|''m'' + 1}}}}.

Comparing the coefficient of {{mvar|j}} in the two expressions of Bernoulli polynomials, one has:

: <math> B_k=\sum_{m=0}^{k-1} (-1)^m \frac{m!}{m+1} S(k-1,m)</math>

(resulting in {{math|''B''<sub>1</sub> {{=}} +{{sfrac|1|2}}}}) which is an explicit formula for Bernoulli numbers and can be used to prove [[Von Staudt–Clausen theorem|Von-Staudt Clausen theorem]].{{r|Boole1880|Gould1972|Apostol2010_197}}

=== 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]]).

===Connection with Pascal's triangle===
There are formulas connecting Pascal's triangle to Bernoulli numbers{{efn|this formula was discovered (or perhaps rediscovered) by Giorgio Pietrocola. His demonstration is available in Italian language {{harv|Pietrocola|2008}}.}}
:<math> B^{+}_n=\frac{|A_n|}{(n+1)!}~~~</math>
where <math>|A_n|</math> is the determinant of a n-by-n [[Hessenberg matrix]] part of [[Pascal's triangle]] whose elements are: <math>
a_{i, k} = \begin{cases} 
0 & \text{if } k>1+i \\
{i+1 \choose k-1} & \text{otherwise}
\end{cases}
</math>

Example:

:<math> B^{+}_6 =\frac{\det\begin{pmatrix}
1& 2& 0& 0& 0& 0\\
1& 3& 3& 0& 0& 0\\
1& 4& 6& 4& 0& 0\\
1& 5& 10& 10& 5& 0\\
1& 6& 15& 20& 15& 6\\
1& 7& 21& 35& 35& 21
\end{pmatrix}}{7!}=\frac{120}{5040}=\frac 1 {42}
</math>

=== Connection with Eulerian numbers ===
There are formulas connecting [[Eulerian number]]s {{math|<big><big>⟨</big></big>{{su|p=''n''|b=''m''|a=c}}<big><big>⟩</big></big>}} to Bernoulli numbers:

:<math>\begin{align}
\sum_{m=0}^n (-1)^m \left \langle {n\atop m} \right \rangle &= 2^{n+1} (2^{n+1}-1) \frac{B_{n+1}}{n+1}, \\
\sum_{m=0}^n (-1)^m \left \langle {n\atop m} \right \rangle \binom{n}{m}^{-1} &= (n+1) B_n.
\end{align}</math>

Both formulae are valid for {{math|''n'' ≥ 0}} if {{math|''B''<sub>1</sub>}} is set to {{sfrac|1|2}}. If {{math|''B''<sub>1</sub>}} is set to −{{sfrac|1|2}} they are valid only for {{math|''n'' ≥ 1}} and {{math|''n'' ≥ 2}} respectively.