@prefix psr: <http://data.loterre.fr/ark:/67375/PSR> .
@prefix skos: <http://www.w3.org/2004/02/skos/core#> .
@prefix dc: <http://purl.org/dc/terms/> .
@prefix xsd: <http://www.w3.org/2001/XMLSchema#> .

psr:-B373Q2P1-V
  skos:prefLabel "combinatorics"@en, "combinatoire"@fr ;
  a skos:Concept ;
  skos:narrower psr:-GTLRXKFD-9 .

psr: a skos:ConceptScheme .
psr:-GTLRXKFD-9
  dc:created "2023-08-03"^^xsd:date ;
  a skos:Concept ;
  skos:exactMatch <https://fr.wikipedia.org/wiki/Nombre_de_Bernoulli>, <https://en.wikipedia.org/wiki/Bernoulli_number> ;
  dc:modified "2023-09-22"^^xsd:date ;
  skos:definition """In mathematics, the Bernoulli numbers Bn are a sequence of rational numbers which occur frequently in analysis. The Bernoulli numbers appear in (and can be defined by) the Taylor series expansions of the tangent and hyperbolic tangent functions, in Faulhaber's formula for the sum of m-th powers of the first n positive integers, in the Euler–Maclaurin formula, and in expressions for certain values of the Riemann zeta function. 
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Bernoulli_number">https://en.wikipedia.org/wiki/Bernoulli_number</a>)"""@en, """En mathématiques, les <b>nombres de Bernoulli</b>, notés <span class="texhtml mvar" style="font-style:italic;">B<sub>n</sub></span> (ou parfois <span class="texhtml mvar" style="font-style:italic;">b<sub>n</sub></span> pour ne pas les confondre avec les polynômes de Bernoulli ou avec les nombres de Bell), constituent une suite de nombres rationnels.
<br/>Ces nombres ont d'abord été étudiés par Jacques Bernoulli (ce qui a conduit Abraham de Moivre à leur donner le nom que nous connaissons aujourd'hui) en cherchant des formules pour exprimer les sommes du type
<br/>
<br/><center><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle \\\\sum _{k=0}^{n-1}k^{m}=0^{m}+1^{m}+2^{m}+\\\\cdots +{(n-1)}^{m}.}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <munderover>
<br/>          <mo>∑<!-- ∑ --></mo>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>k</mi>
<br/>            <mo>=</mo>
<br/>            <mn>0</mn>
<br/>          </mrow>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>n</mi>
<br/>            <mo>−<!-- − --></mo>
<br/>            <mn>1</mn>
<br/>          </mrow>
<br/>        </munderover>
<br/>        <msup>
<br/>          <mi>k</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>m</mi>
<br/>          </mrow>
<br/>        </msup>
<br/>        <mo>=</mo>
<br/>        <msup>
<br/>          <mn>0</mn>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>m</mi>
<br/>          </mrow>
<br/>        </msup>
<br/>        <mo>+</mo>
<br/>        <msup>
<br/>          <mn>1</mn>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>m</mi>
<br/>          </mrow>
<br/>        </msup>
<br/>        <mo>+</mo>
<br/>        <msup>
<br/>          <mn>2</mn>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>m</mi>
<br/>          </mrow>
<br/>        </msup>
<br/>        <mo>+</mo>
<br/>        <mo>⋯<!-- ⋯ --></mo>
<br/>        <mo>+</mo>
<br/>        <msup>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mo stretchy="false">(</mo>
<br/>            <mi>n</mi>
<br/>            <mo>−<!-- − --></mo>
<br/>            <mn>1</mn>
<br/>            <mo stretchy="false">)</mo>
<br/>          </mrow>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>m</mi>
<br/>          </mrow>
<br/>        </msup>
<br/>        <mo>.</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle \\\\sum _{k=0}^{n-1}k^{m}=0^{m}+1^{m}+2^{m}+\\\\cdots +{(n-1)}^{m}.}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d79ec6de0ba7686e7d1ac9005a1b8ea23f62bb10" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -3.171ex; width:41.853ex; height:7.509ex;" alt="{\\\\displaystyle \\\\sum _{k=0}^{n-1}k^{m}=0^{m}+1^{m}+2^{m}+\\\\cdots +{(n-1)}^{m}.}"></span></center>
<br/>Pour des valeurs entières de <span class="texhtml mvar" style="font-style:italic;">m</span>, cette somme s'écrit comme un polynôme de la variable <i>n</i> dont les premiers termes sont&nbsp;:
<br/>
<br/><center><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle \\\\sum _{k=0}^{n-1}k^{m}={\\rac {1}{m+1}}\\\\left(n^{m+1}-{\\rac {1}{2}}{m+1 \\\\choose 1}{n^{m}}+{\\rac {1}{6}}{m+1 \\\\choose 2}{n^{m-1}}-{\\rac {1}{30}}{m+1 \\\\choose 4}{n^{m-3}}+{\\rac {1}{42}}{m+1 \\\\choose 6}{n^{m-5}}+\\\\ldots \\
ight).}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <munderover>
<br/>          <mo>∑<!-- ∑ --></mo>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>k</mi>
<br/>            <mo>=</mo>
<br/>            <mn>0</mn>
<br/>          </mrow>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>n</mi>
<br/>            <mo>−<!-- − --></mo>
<br/>            <mn>1</mn>
<br/>          </mrow>
<br/>        </munderover>
<br/>        <msup>
<br/>          <mi>k</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>m</mi>
<br/>          </mrow>
<br/>        </msup>
<br/>        <mo>=</mo>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mfrac>
<br/>            <mn>1</mn>
<br/>            <mrow>
<br/>              <mi>m</mi>
<br/>              <mo>+</mo>
<br/>              <mn>1</mn>
<br/>            </mrow>
<br/>          </mfrac>
<br/>        </mrow>
<br/>        <mrow>
<br/>          <mo>(</mo>
<br/>          <mrow>
<br/>            <msup>
<br/>              <mi>n</mi>
<br/>              <mrow class="MJX-TeXAtom-ORD">
<br/>                <mi>m</mi>
<br/>                <mo>+</mo>
<br/>                <mn>1</mn>
<br/>              </mrow>
<br/>            </msup>
<br/>            <mo>−<!-- − --></mo>
<br/>            <mrow class="MJX-TeXAtom-ORD">
<br/>              <mfrac>
<br/>                <mn>1</mn>
<br/>                <mn>2</mn>
<br/>              </mfrac>
<br/>            </mrow>
<br/>            <mrow class="MJX-TeXAtom-ORD">
<br/>              <mrow>
<br/>                <mrow class="MJX-TeXAtom-OPEN">
<br/>                  <mo maxsize="2.047em" minsize="2.047em">(</mo>
<br/>                </mrow>
<br/>                <mfrac linethickness="0">
<br/>                  <mrow>
<br/>                    <mi>m</mi>
<br/>                    <mo>+</mo>
<br/>                    <mn>1</mn>
<br/>                  </mrow>
<br/>                  <mn>1</mn>
<br/>                </mfrac>
<br/>                <mrow class="MJX-TeXAtom-CLOSE">
<br/>                  <mo maxsize="2.047em" minsize="2.047em">)</mo>
<br/>                </mrow>
<br/>              </mrow>
<br/>            </mrow>
<br/>            <mrow class="MJX-TeXAtom-ORD">
<br/>              <msup>
<br/>                <mi>n</mi>
<br/>                <mrow class="MJX-TeXAtom-ORD">
<br/>                  <mi>m</mi>
<br/>                </mrow>
<br/>              </msup>
<br/>            </mrow>
<br/>            <mo>+</mo>
<br/>            <mrow class="MJX-TeXAtom-ORD">
<br/>              <mfrac>
<br/>                <mn>1</mn>
<br/>                <mn>6</mn>
<br/>              </mfrac>
<br/>            </mrow>
<br/>            <mrow class="MJX-TeXAtom-ORD">
<br/>              <mrow>
<br/>                <mrow class="MJX-TeXAtom-OPEN">
<br/>                  <mo maxsize="2.047em" minsize="2.047em">(</mo>
<br/>                </mrow>
<br/>                <mfrac linethickness="0">
<br/>                  <mrow>
<br/>                    <mi>m</mi>
<br/>                    <mo>+</mo>
<br/>                    <mn>1</mn>
<br/>                  </mrow>
<br/>                  <mn>2</mn>
<br/>                </mfrac>
<br/>                <mrow class="MJX-TeXAtom-CLOSE">
<br/>                  <mo maxsize="2.047em" minsize="2.047em">)</mo>
<br/>                </mrow>
<br/>              </mrow>
<br/>            </mrow>
<br/>            <mrow class="MJX-TeXAtom-ORD">
<br/>              <msup>
<br/>                <mi>n</mi>
<br/>                <mrow class="MJX-TeXAtom-ORD">
<br/>                  <mi>m</mi>
<br/>                  <mo>−<!-- − --></mo>
<br/>                  <mn>1</mn>
<br/>                </mrow>
<br/>              </msup>
<br/>            </mrow>
<br/>            <mo>−<!-- − --></mo>
<br/>            <mrow class="MJX-TeXAtom-ORD">
<br/>              <mfrac>
<br/>                <mn>1</mn>
<br/>                <mn>30</mn>
<br/>              </mfrac>
<br/>            </mrow>
<br/>            <mrow class="MJX-TeXAtom-ORD">
<br/>              <mrow>
<br/>                <mrow class="MJX-TeXAtom-OPEN">
<br/>                  <mo maxsize="2.047em" minsize="2.047em">(</mo>
<br/>                </mrow>
<br/>                <mfrac linethickness="0">
<br/>                  <mrow>
<br/>                    <mi>m</mi>
<br/>                    <mo>+</mo>
<br/>                    <mn>1</mn>
<br/>                  </mrow>
<br/>                  <mn>4</mn>
<br/>                </mfrac>
<br/>                <mrow class="MJX-TeXAtom-CLOSE">
<br/>                  <mo maxsize="2.047em" minsize="2.047em">)</mo>
<br/>                </mrow>
<br/>              </mrow>
<br/>            </mrow>
<br/>            <mrow class="MJX-TeXAtom-ORD">
<br/>              <msup>
<br/>                <mi>n</mi>
<br/>                <mrow class="MJX-TeXAtom-ORD">
<br/>                  <mi>m</mi>
<br/>                  <mo>−<!-- − --></mo>
<br/>                  <mn>3</mn>
<br/>                </mrow>
<br/>              </msup>
<br/>            </mrow>
<br/>            <mo>+</mo>
<br/>            <mrow class="MJX-TeXAtom-ORD">
<br/>              <mfrac>
<br/>                <mn>1</mn>
<br/>                <mn>42</mn>
<br/>              </mfrac>
<br/>            </mrow>
<br/>            <mrow class="MJX-TeXAtom-ORD">
<br/>              <mrow>
<br/>                <mrow class="MJX-TeXAtom-OPEN">
<br/>                  <mo maxsize="2.047em" minsize="2.047em">(</mo>
<br/>                </mrow>
<br/>                <mfrac linethickness="0">
<br/>                  <mrow>
<br/>                    <mi>m</mi>
<br/>                    <mo>+</mo>
<br/>                    <mn>1</mn>
<br/>                  </mrow>
<br/>                  <mn>6</mn>
<br/>                </mfrac>
<br/>                <mrow class="MJX-TeXAtom-CLOSE">
<br/>                  <mo maxsize="2.047em" minsize="2.047em">)</mo>
<br/>                </mrow>
<br/>              </mrow>
<br/>            </mrow>
<br/>            <mrow class="MJX-TeXAtom-ORD">
<br/>              <msup>
<br/>                <mi>n</mi>
<br/>                <mrow class="MJX-TeXAtom-ORD">
<br/>                  <mi>m</mi>
<br/>                  <mo>−<!-- − --></mo>
<br/>                  <mn>5</mn>
<br/>                </mrow>
<br/>              </msup>
<br/>            </mrow>
<br/>            <mo>+</mo>
<br/>            <mo>…<!-- … --></mo>
<br/>          </mrow>
<br/>          <mo>)</mo>
<br/>        </mrow>
<br/>        <mo>.</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle \\\\sum _{k=0}^{n-1}k^{m}={\\rac {1}{m+1}}\\\\left(n^{m+1}-{\\rac {1}{2}}{m+1 \\\\choose 1}{n^{m}}+{\\rac {1}{6}}{m+1 \\\\choose 2}{n^{m-1}}-{\\rac {1}{30}}{m+1 \\\\choose 4}{n^{m-3}}+{\\rac {1}{42}}{m+1 \\\\choose 6}{n^{m-5}}+\\\\ldots \\
ight).}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81515120a5bb65fe6d5a798dd8906f4cb83cccf8" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -3.171ex; width:110.301ex; height:7.509ex;" alt="\\\\sum _{{k=0}}^{{n-1}}k^{m}={\\rac  1{m+1}}\\\\left(n^{{m+1}}-{\\rac  12}{m+1 \\\\choose 1}{n^{m}}+{\\rac  16}{m+1 \\\\choose 2}{n^{{m-1}}}-{\\rac  1{30}}{m+1 \\\\choose 4}{n^{{m-3}}}+{\\rac  1{42}}{m+1 \\\\choose 6}{n^{{m-5}}}+\\\\ldots \\
ight)."> </center>
<br/>(Wikipedia, L'Encylopédie Libre, <a href="https://fr.wikipedia.org/wiki/Nombre_de_Bernoulli">https://fr.wikipedia.org/wiki/Nombre_de_Bernoulli</a>)"""@fr ;
  skos:inScheme psr: ;
  skos:broader psr:-B373Q2P1-V, psr:-FM1M1PDT-5 ;
  skos:prefLabel "Bernoulli number"@en, "nombre de Bernoulli"@fr .

psr:-FM1M1PDT-5
  skos:prefLabel "suite d'entiers"@fr, "integer sequence"@en ;
  a skos:Concept ;
  skos:narrower psr:-GTLRXKFD-9 .