@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:-WM7JPRRL-8
  skos:prefLabel "loi binomiale négative"@fr, "negative binomial distribution"@en ;
  a skos:Concept ;
  skos:related psr:-F7H3K8H1-0 .

psr:-CZ3HC78N-1
  skos:prefLabel "identité de Dixon"@fr, "Dixon's identity"@en ;
  a skos:Concept ;
  skos:broader psr:-F7H3K8H1-0 .

psr:-QKQ1ZP6D-T
  skos:prefLabel "Fuss-Catalan number"@en, "nombre de Fuss-Catalan"@fr ;
  a skos:Concept ;
  skos:broader psr:-F7H3K8H1-0 .

psr:-G8L5RZH2-4
  skos:prefLabel "binomial theorem"@en, "formule du binôme de Newton"@fr ;
  a skos:Concept ;
  skos:related psr:-F7H3K8H1-0 .

psr:-RH5QPFJR-5
  skos:prefLabel "triangle de Pascal"@fr, "Pascal's triangle"@en ;
  a skos:Concept ;
  skos:broader psr:-F7H3K8H1-0 .

psr:-N05ZGM7Q-N
  skos:prefLabel "Catalan number"@en, "nombre de Catalan"@fr ;
  a skos:Concept ;
  skos:broader psr:-F7H3K8H1-0 .

psr:-TF44DDMB-4
  skos:prefLabel "central binomial coefficient"@en, "coefficient binomial central"@fr ;
  a skos:Concept ;
  skos:broader psr:-F7H3K8H1-0 .

psr:-W24S9MNF-2
  skos:prefLabel "Bernoulli's triangle"@en, "triangle de Bernoulli"@fr ;
  a skos:Concept ;
  skos:broader psr:-F7H3K8H1-0 .

psr: a skos:ConceptScheme .
psr:-DGVD7PPH-S
  skos:prefLabel "Abel's binomial theorem"@en, "théorème binomial d'Abel"@fr ;
  a skos:Concept ;
  skos:broader psr:-F7H3K8H1-0 .

psr:-B373Q2P1-V
  skos:prefLabel "combinatorics"@en, "combinatoire"@fr ;
  a skos:Concept ;
  skos:narrower psr:-F7H3K8H1-0 .

psr:-F7H3K8H1-0
  skos:related psr:-G8L5RZH2-4, psr:-WM7JPRRL-8 ;
  skos:altLabel "coefficient du binôme"@fr ;
  skos:definition """En mathématiques, les <b>coefficients binomiaux</b>, ou <b>coefficients du binôme</b>, définis pour tout entier naturel <span class="texhtml mvar" style="font-style:italic;">n</span> et tout entier naturel <span class="texhtml mvar" style="font-style:italic;">k</span> inférieur ou égal à <span class="texhtml mvar" style="font-style:italic;">n</span>, donnent le nombre de parties à <span class="texhtml mvar" style="font-style:italic;">k</span> éléments d'un ensemble à <span class="texhtml mvar" style="font-style:italic;">n</span> éléments. On les note <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 \\	extstyle {n \\\\choose k}}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mstyle displaystyle="false" scriptlevel="0">
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mrow>
<br/>              <mrow class="MJX-TeXAtom-OPEN">
<br/>                <mo maxsize="1.2em" minsize="1.2em">(</mo>
<br/>              </mrow>
<br/>              <mfrac linethickness="0">
<br/>                <mi>n</mi>
<br/>                <mi>k</mi>
<br/>              </mfrac>
<br/>              <mrow class="MJX-TeXAtom-CLOSE">
<br/>                <mo maxsize="1.2em" minsize="1.2em">)</mo>
<br/>              </mrow>
<br/>            </mrow>
<br/>          </mrow>
<br/>        </mstyle>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle \\	extstyle {n \\\\choose k}}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af96754948220f504c1e912a8fb6dc1bbc2b1ba4" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -1.005ex; width:3.116ex; height:3.176ex;" alt="\\	extstyle {n \\\\choose k}"></span>  - qui se lit «&nbsp;<span class="texhtml mvar" style="font-style:italic;">k</span> parmi <span class="texhtml mvar" style="font-style:italic;">n</span>&nbsp;»  - ou <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 \\	extstyle {C_{n}^{k}}}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mstyle displaystyle="false" scriptlevel="0">
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <msubsup>
<br/>              <mi>C</mi>
<br/>              <mrow class="MJX-TeXAtom-ORD">
<br/>                <mi>n</mi>
<br/>              </mrow>
<br/>              <mrow class="MJX-TeXAtom-ORD">
<br/>                <mi>k</mi>
<br/>              </mrow>
<br/>            </msubsup>
<br/>          </mrow>
<br/>        </mstyle>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle \\	extstyle {C_{n}^{k}}}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de8af0cb26591c3b42d7a1168668894438bae833" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:2.887ex; height:2.843ex;" alt="{\\\\displaystyle \\	extstyle {C_{n}^{k}}}"></span>, la lettre C étant l'initiale du mot «&nbsp;combinaison&nbsp;»
<br/>Les coefficients binomiaux s'expriment à l'aide de la fonction factorielle&nbsp;:
<br/>
<br/><dl><dd><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 {n \\\\choose k}=C_{n}^{k}\\\\,={\\rac {n!}{k!(n-k)!}}}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<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/>              <mi>n</mi>
<br/>              <mi>k</mi>
<br/>            </mfrac>
<br/>            <mrow class="MJX-TeXAtom-CLOSE">
<br/>              <mo maxsize="2.047em" minsize="2.047em">)</mo>
<br/>            </mrow>
<br/>          </mrow>
<br/>        </mrow>
<br/>        <mo>=</mo>
<br/>        <msubsup>
<br/>          <mi>C</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>n</mi>
<br/>          </mrow>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>k</mi>
<br/>          </mrow>
<br/>        </msubsup>
<br/>        <mspace width="thinmathspace"></mspace>
<br/>        <mo>=</mo>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mfrac>
<br/>            <mrow>
<br/>              <mi>n</mi>
<br/>              <mo>!</mo>
<br/>            </mrow>
<br/>            <mrow>
<br/>              <mi>k</mi>
<br/>              <mo>!</mo>
<br/>              <mo stretchy="false">(</mo>
<br/>              <mi>n</mi>
<br/>              <mo>−<!-- − --></mo>
<br/>              <mi>k</mi>
<br/>              <mo stretchy="false">)</mo>
<br/>              <mo>!</mo>
<br/>            </mrow>
<br/>          </mfrac>
<br/>        </mrow>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle {n \\\\choose k}=C_{n}^{k}\\\\,={\\rac {n!}{k!(n-k)!}}}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8678a3f02f0cb9c4bb5c013bdc3a8836ad1d5be9" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.671ex; width:24.883ex; height:6.343ex;" alt="{n \\\\choose k}=C_{n}^{k}\\\\,={\\rac  {n!}{k!(n-k)!}}"></span>.</dd></dl>
<br/>Ils interviennent dans de nombreux domaines des mathématiques&nbsp;: développement du binôme en algèbre, dénombrements, développement en série, lois de probabilités, etc. On peut les généraliser, sous certaines conditions, aux nombres complexes. 
<br/>(Wikipedia, L'Encylopédie Libre, <a href="https://fr.wikipedia.org/wiki/Coefficient_binomial">https://fr.wikipedia.org/wiki/Coefficient_binomial</a>)"""@fr, """In mathematics, the <b>binomial coefficients</b> are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers <span class="texhtml"><i>n</i> ≥ <i>k</i> ≥ 0</span> and is written <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 {\\	binom {n}{k}}.}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mstyle displaystyle="false" scriptlevel="0">
<br/>            <mrow>
<br/>              <mrow class="MJX-TeXAtom-OPEN">
<br/>                <mo maxsize="1.2em" minsize="1.2em">(</mo>
<br/>              </mrow>
<br/>              <mfrac linethickness="0">
<br/>                <mi>n</mi>
<br/>                <mi>k</mi>
<br/>              </mfrac>
<br/>              <mrow class="MJX-TeXAtom-CLOSE">
<br/>                <mo maxsize="1.2em" minsize="1.2em">)</mo>
<br/>              </mrow>
<br/>            </mrow>
<br/>          </mstyle>
<br/>        </mrow>
<br/>        <mo>.</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle {\\	binom {n}{k}}.}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39c7822635a8fa426d00ca72733ea1bd6fe90b01" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -1.005ex; width:3.763ex; height:3.176ex;" alt="{\\\\displaystyle {\\	binom {n}{k}}.}"></span> It is the coefficient of the <span class="texhtml"><i>x</i><sup><i>k</i></sup></span> term in the polynomial expansion of the binomial power <span class="texhtml">(1 + <i>x</i>)<sup><i>n</i></sup></span>; this coefficient can be computed by the multiplicative formula
<br/>
<br/><dl><dd><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 {\\inom {n}{k}}={\\rac {n\\	imes (n-1)\\	imes \\\\cdots \\	imes (n-k+1)}{k\\	imes (k-1)\\	imes \\\\cdots \\	imes 1}},}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<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/>              <mi>n</mi>
<br/>              <mi>k</mi>
<br/>            </mfrac>
<br/>            <mrow class="MJX-TeXAtom-CLOSE">
<br/>              <mo maxsize="2.047em" minsize="2.047em">)</mo>
<br/>            </mrow>
<br/>          </mrow>
<br/>        </mrow>
<br/>        <mo>=</mo>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mfrac>
<br/>            <mrow>
<br/>              <mi>n</mi>
<br/>              <mo>×<!-- × --></mo>
<br/>              <mo stretchy="false">(</mo>
<br/>              <mi>n</mi>
<br/>              <mo>−<!-- − --></mo>
<br/>              <mn>1</mn>
<br/>              <mo stretchy="false">)</mo>
<br/>              <mo>×<!-- × --></mo>
<br/>              <mo>⋯<!-- ⋯ --></mo>
<br/>              <mo>×<!-- × --></mo>
<br/>              <mo stretchy="false">(</mo>
<br/>              <mi>n</mi>
<br/>              <mo>−<!-- − --></mo>
<br/>              <mi>k</mi>
<br/>              <mo>+</mo>
<br/>              <mn>1</mn>
<br/>              <mo stretchy="false">)</mo>
<br/>            </mrow>
<br/>            <mrow>
<br/>              <mi>k</mi>
<br/>              <mo>×<!-- × --></mo>
<br/>              <mo stretchy="false">(</mo>
<br/>              <mi>k</mi>
<br/>              <mo>−<!-- − --></mo>
<br/>              <mn>1</mn>
<br/>              <mo stretchy="false">)</mo>
<br/>              <mo>×<!-- × --></mo>
<br/>              <mo>⋯<!-- ⋯ --></mo>
<br/>              <mo>×<!-- × --></mo>
<br/>              <mn>1</mn>
<br/>            </mrow>
<br/>          </mfrac>
<br/>        </mrow>
<br/>        <mo>,</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle {\\inom {n}{k}}={\\rac {n\\	imes (n-1)\\	imes \\\\cdots \\	imes (n-k+1)}{k\\	imes (k-1)\\	imes \\\\cdots \\	imes 1}},}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7c293b21bed306b0c914b842f3a04aafc78a88c" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.671ex; width:40.502ex; height:6.509ex;" alt="{\\\\displaystyle {\\inom {n}{k}}={\\rac {n\\	imes (n-1)\\	imes \\\\cdots \\	imes (n-k+1)}{k\\	imes (k-1)\\	imes \\\\cdots \\	imes 1}},}"></span></dd></dl>
<br/>which using factorial notation can be compactly expressed as
<br/>
<br/><dl><dd><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 {\\inom {n}{k}}={\\rac {n!}{k!(n-k)!}}.}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<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/>              <mi>n</mi>
<br/>              <mi>k</mi>
<br/>            </mfrac>
<br/>            <mrow class="MJX-TeXAtom-CLOSE">
<br/>              <mo maxsize="2.047em" minsize="2.047em">)</mo>
<br/>            </mrow>
<br/>          </mrow>
<br/>        </mrow>
<br/>        <mo>=</mo>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mfrac>
<br/>            <mrow>
<br/>              <mi>n</mi>
<br/>              <mo>!</mo>
<br/>            </mrow>
<br/>            <mrow>
<br/>              <mi>k</mi>
<br/>              <mo>!</mo>
<br/>              <mo stretchy="false">(</mo>
<br/>              <mi>n</mi>
<br/>              <mo>−<!-- − --></mo>
<br/>              <mi>k</mi>
<br/>              <mo stretchy="false">)</mo>
<br/>              <mo>!</mo>
<br/>            </mrow>
<br/>          </mfrac>
<br/>        </mrow>
<br/>        <mo>.</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle {\\inom {n}{k}}={\\rac {n!}{k!(n-k)!}}.}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a2457a7ef3c77831e34e06a1fe17a80b84a03181" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.671ex; width:19.158ex; height:6.343ex;" alt="{\\\\displaystyle {\\inom {n}{k}}={\\rac {n!}{k!(n-k)!}}.}"> 
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Binomial_coefficient">https://en.wikipedia.org/wiki/Binomial_coefficient</a>)"""@en ;
  skos:narrower psr:-QKQ1ZP6D-T, psr:-N05ZGM7Q-N, psr:-TF44DDMB-4, psr:-W24S9MNF-2, psr:-DGVD7PPH-S, psr:-RH5QPFJR-5, psr:-CZ3HC78N-1 ;
  dc:modified "2023-08-24"^^xsd:date ;
  skos:broader psr:-FM1M1PDT-5, psr:-B373Q2P1-V ;
  skos:exactMatch <https://fr.wikipedia.org/wiki/Coefficient_binomial>, <https://en.wikipedia.org/wiki/Binomial_coefficient> ;
  skos:inScheme psr: ;
  skos:prefLabel "coefficient binomial"@fr, "binomial coefficient"@en ;
  a skos:Concept .

psr:-FM1M1PDT-5
  skos:prefLabel "suite d'entiers"@fr, "integer sequence"@en ;
  a skos:Concept ;
  skos:narrower psr:-F7H3K8H1-0 .