@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:-TF44DDMB-4
  skos:broader psr:-F7H3K8H1-0 ;
  dc:modified "2024-10-18"^^xsd:date ;
  skos:exactMatch <https://en.wikipedia.org/wiki/Central_binomial_coefficient>, <https://fr.wikipedia.org/wiki/Coefficient_binomial_central> ;
  a skos:Concept ;
  skos:prefLabel "central binomial coefficient"@en, "coefficient binomial central"@fr ;
  skos:definition """En mathématiques le <b>coefficient binomial central</b> d'ordre <span class="texhtml mvar" style="font-style:italic;">n</span> est le coefficient binomial défini par :  <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 {\\\\dbinom {2n}{n}}={\\rac {(2n)!}{(n!)^{2}}}=\\\\prod \\\\limits _{k=1}^{n}{\\rac {n+k}{k}}{\\	ext{ pour tout }}n\\\\geqslant 0.}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mrow class="MJX-TeXAtom-ORD">           <mstyle displaystyle="true" scriptlevel="0">             <mrow>               <mrow class="MJX-TeXAtom-OPEN">                 <mo maxsize="2.047em" minsize="2.047em">(</mo>               </mrow>               <mfrac linethickness="0">                 <mrow>                   <mn>2</mn>                   <mi>n</mi>                 </mrow>                 <mi>n</mi>               </mfrac>               <mrow class="MJX-TeXAtom-CLOSE">                 <mo maxsize="2.047em" minsize="2.047em">)</mo>               </mrow>             </mrow>           </mstyle>         </mrow>         <mo>=</mo>         <mrow class="MJX-TeXAtom-ORD">           <mfrac>             <mrow>               <mo stretchy="false">(</mo>               <mn>2</mn>               <mi>n</mi>               <mo stretchy="false">)</mo>               <mo>!</mo>             </mrow>             <mrow>               <mo stretchy="false">(</mo>               <mi>n</mi>               <mo>!</mo>               <msup>                 <mo stretchy="false">)</mo>                 <mrow class="MJX-TeXAtom-ORD">                   <mn>2</mn>                 </mrow>               </msup>             </mrow>           </mfrac>         </mrow>         <mo>=</mo>         <munderover>           <mo movablelimits="false">∏<!-- ∏ --></mo>           <mrow class="MJX-TeXAtom-ORD">             <mi>k</mi>             <mo>=</mo>             <mn>1</mn>           </mrow>           <mrow class="MJX-TeXAtom-ORD">             <mi>n</mi>           </mrow>         </munderover>         <mrow class="MJX-TeXAtom-ORD">           <mfrac>             <mrow>               <mi>n</mi>               <mo>+</mo>               <mi>k</mi>             </mrow>             <mi>k</mi>           </mfrac>         </mrow>         <mrow class="MJX-TeXAtom-ORD">           <mtext> pour tout </mtext>         </mrow>         <mi>n</mi>         <mo>⩾<!-- ⩾ --></mo>         <mn>0.</mn>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle {\\\\dbinom {2n}{n}}={\\rac {(2n)!}{(n!)^{2}}}=\\\\prod \\\\limits _{k=1}^{n}{\\rac {n+k}{k}}{\\	ext{ pour tout }}n\\\\geqslant 0.}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bfc60fd7f034b9b1dd5a8e6545f2c68527edbf1c" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:44.632ex; height:6.843ex;" alt="{\\\\displaystyle {\\\\dbinom {2n}{n}}={\\rac {(2n)!}{(n!)^{2}}}=\\\\prod \\\\limits _{k=1}^{n}{\\rac {n+k}{k}}{\\	ext{ pour tout }}n\\\\geqslant 0.}"></span></dd></dl> Il est ainsi nommé pour la position centrale qu'il occupe dans la liste des <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 {\\\\dbinom {2n}{k}}}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mrow class="MJX-TeXAtom-ORD">           <mstyle displaystyle="true" scriptlevel="0">             <mrow>               <mrow class="MJX-TeXAtom-OPEN">                 <mo maxsize="2.047em" minsize="2.047em">(</mo>               </mrow>               <mfrac linethickness="0">                 <mrow>                   <mn>2</mn>                   <mi>n</mi>                 </mrow>                 <mi>k</mi>               </mfrac>               <mrow class="MJX-TeXAtom-CLOSE">                 <mo maxsize="2.047em" minsize="2.047em">)</mo>               </mrow>             </mrow>           </mstyle>         </mrow>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle {\\\\dbinom {2n}{k}}}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92612c77ffbb30f6e392da2403307b1b1d06a432" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:5.978ex; height:6.176ex;" alt="{\\\\displaystyle {\\\\dbinom {2n}{k}}}"></span> pour <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 0\\\\leqslant k\\\\leqslant 2n}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mn>0</mn>         <mo>⩽<!-- ⩽ --></mo>         <mi>k</mi>         <mo>⩽<!-- ⩽ --></mo>         <mn>2</mn>         <mi>n</mi>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle 0\\\\leqslant k\\\\leqslant 2n}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fcdaf5b9246a02d338627703dbd4742c03376b3f" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.128ex; height:2.343ex;" alt="{\\\\displaystyle 0\\\\leqslant k\\\\leqslant 2n}"></span> (ligne d'indice <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 2n}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mn>2</mn>         <mi>n</mi>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle 2n}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/134afa8ff09fdddd24b06f289e92e3a045092bd1" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.557ex; height:2.176ex;" alt="2n"></span> du triangle de Pascal) ; l'identité de Vandermonde : <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 {2n}{n}}=\\\\sum _{k=0}^{n}{\\inom {n}{k}}^{2}}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mrow class="MJX-TeXAtom-ORD">           <mrow>             <mrow class="MJX-TeXAtom-OPEN">               <mo maxsize="2.047em" minsize="2.047em">(</mo>             </mrow>             <mfrac linethickness="0">               <mrow>                 <mn>2</mn>                 <mi>n</mi>               </mrow>               <mi>n</mi>             </mfrac>             <mrow class="MJX-TeXAtom-CLOSE">               <mo maxsize="2.047em" minsize="2.047em">)</mo>             </mrow>           </mrow>         </mrow>         <mo>=</mo>         <munderover>           <mo>∑<!-- ∑ --></mo>           <mrow class="MJX-TeXAtom-ORD">             <mi>k</mi>             <mo>=</mo>             <mn>0</mn>           </mrow>           <mrow class="MJX-TeXAtom-ORD">             <mi>n</mi>           </mrow>         </munderover>         <msup>           <mrow class="MJX-TeXAtom-ORD">             <mrow>               <mrow class="MJX-TeXAtom-OPEN">                 <mo maxsize="2.047em" minsize="2.047em">(</mo>               </mrow>               <mfrac linethickness="0">                 <mi>n</mi>                 <mi>k</mi>               </mfrac>               <mrow class="MJX-TeXAtom-CLOSE">                 <mo maxsize="2.047em" minsize="2.047em">)</mo>               </mrow>             </mrow>           </mrow>           <mrow class="MJX-TeXAtom-ORD">             <mn>2</mn>           </mrow>         </msup>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle {\\inom {2n}{n}}=\\\\sum _{k=0}^{n}{\\inom {n}{k}}^{2}}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/228e18e09b46a60da1013926947adee734ad5a48" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:18.689ex; height:7.176ex;" alt="{\\\\displaystyle {\\inom {2n}{n}}=\\\\sum _{k=0}^{n}{\\inom {n}{k}}^{2}}"></span>montre qu'il s'obtient par la somme des carrés des termes de la ligne d'indice <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}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi>n</mi>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle n}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="n"></span> de ce triangle. Pour les premières valeurs de <span class="texhtml mvar" style="font-style:italic;">n</span>, celles du coefficient binomial central associé sont : 1, 2, 6, 20, 70, 252. La liste de toutes les valeurs constitue la suite A000984 de l'OEIS. 
<br/>(Wikipedia, L'Encylopédie Libre, <a href="https://fr.wikipedia.org/wiki/Coefficient_binomial_central">https://fr.wikipedia.org/wiki/Coefficient_binomial_central</a>)"""@fr, """In mathematics the <i>n</i>th <b>central binomial coefficient</b> is the particular binomial coefficient  <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 {2n \\\\choose n}={\\rac {(2n)!}{(n!)^{2}}}=\\\\prod \\\\limits _{k=1}^{n}{\\rac {n+k}{k}}{\\	ext{ for all }}n\\\\geq 0.}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mrow class="MJX-TeXAtom-ORD">           <mrow>             <mrow class="MJX-TeXAtom-OPEN">               <mo maxsize="2.047em" minsize="2.047em">(</mo>             </mrow>             <mfrac linethickness="0">               <mrow>                 <mn>2</mn>                 <mi>n</mi>               </mrow>               <mi>n</mi>             </mfrac>             <mrow class="MJX-TeXAtom-CLOSE">               <mo maxsize="2.047em" minsize="2.047em">)</mo>             </mrow>           </mrow>         </mrow>         <mo>=</mo>         <mrow class="MJX-TeXAtom-ORD">           <mfrac>             <mrow>               <mo stretchy="false">(</mo>               <mn>2</mn>               <mi>n</mi>               <mo stretchy="false">)</mo>               <mo>!</mo>             </mrow>             <mrow>               <mo stretchy="false">(</mo>               <mi>n</mi>               <mo>!</mo>               <msup>                 <mo stretchy="false">)</mo>                 <mrow class="MJX-TeXAtom-ORD">                   <mn>2</mn>                 </mrow>               </msup>             </mrow>           </mfrac>         </mrow>         <mo>=</mo>         <munderover>           <mo movablelimits="false">∏<!-- ∏ --></mo>           <mrow class="MJX-TeXAtom-ORD">             <mi>k</mi>             <mo>=</mo>             <mn>1</mn>           </mrow>           <mrow class="MJX-TeXAtom-ORD">             <mi>n</mi>           </mrow>         </munderover>         <mrow class="MJX-TeXAtom-ORD">           <mfrac>             <mrow>               <mi>n</mi>               <mo>+</mo>               <mi>k</mi>             </mrow>             <mi>k</mi>           </mfrac>         </mrow>         <mrow class="MJX-TeXAtom-ORD">           <mtext> for all </mtext>         </mrow>         <mi>n</mi>         <mo>≥<!-- ≥ --></mo>         <mn>0.</mn>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle {2n \\\\choose n}={\\rac {(2n)!}{(n!)^{2}}}=\\\\prod \\\\limits _{k=1}^{n}{\\rac {n+k}{k}}{\\	ext{ for all }}n\\\\geq 0.}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0941f1e66c5c8cf329432d12f759984f9f08d21f" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:40.95ex; height:6.843ex;" alt="{\\\\displaystyle {2n \\\\choose n}={\\rac {(2n)!}{(n!)^{2}}}=\\\\prod \\\\limits _{k=1}^{n}{\\rac {n+k}{k}}{\\	ext{ for all }}n\\\\geq 0.}"></span></dd></dl> They are called central since they show up exactly in the middle of the even-numbered rows in Pascal's triangle. The first few central binomial coefficients starting at <i>n</i> = 0 are:  <dl><dd>1, 2, 6, 20, 70, 252, 924, 3432, 12870, 48620, ...; (sequence A000984 in the OEIS)</dd> 
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Central_binomial_coefficient">https://en.wikipedia.org/wiki/Central_binomial_coefficient</a>)"""@en ;
  dc:created "2023-08-24"^^xsd:date ;
  skos:inScheme psr: .

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

psr: a skos:ConceptScheme .