@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: a skos:ConceptScheme .
psr:-W127WDLN-J
  skos:prefLabel "factorielle"@fr, "factorial"@en ;
  a skos:Concept ;
  skos:narrower psr:-Z3HLMRBS-K .

psr:-Z3HLMRBS-K
  dc:modified "2024-10-18"^^xsd:date ;
  skos:definition """La <b>multifactorielle</b> d'ordre <span class="texhtml"><i>q</i></span> de <span class="texhtml"><i>n</i></span> ne conserve, dans le produit définissant la factorielle, qu'un facteur sur <span class="texhtml"><i>q</i></span>, en partant de <span class="texhtml"><i>n</i></span>. Afin d'alléger l'écriture, une notation courante est d'utiliser <span class="texhtml"><i>q</i></span> points d'exclamation suivant le nombre <span class="texhtml"><i>n</i></span> pour noter cette fonction.  
<br/>(Wikipedia, L'Encylopédie Libre, <a href="https://fr.wikipedia.org/wiki/Analogues_de_la_factorielle#Multifactorielles">https://fr.wikipedia.org/wiki/Analogues_de_la_factorielle#Multifactorielles</a>)"""@fr, """In mathematics, the <b>double factorial</b> of a number <span class="texhtml mvar" style="font-style:italic;">n</span>, denoted by <span class="texhtml"><i>n</i>‼</span>, is the product of all the positive integers up to <span class="texhtml mvar" style="font-style:italic;">n</span> that have the same parity (odd or even) as <span class="texhtml mvar" style="font-style:italic;">n</span>. That is, <div class="mwe-math-element"><div class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle n!!=\\\\prod _{k=0}^{\\\\left\\\\lceil {\\rac {n}{2}}\\
ight\\
ceil -1}(n-2k)=n(n-2)(n-4)\\\\cdots .}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi>n</mi>         <mo>!</mo>         <mo>!</mo>         <mo>=</mo>         <munderover>           <mo>∏<!-- ∏ --></mo>           <mrow class="MJX-TeXAtom-ORD">             <mi>k</mi>             <mo>=</mo>             <mn>0</mn>           </mrow>           <mrow class="MJX-TeXAtom-ORD">             <mrow>               <mo>⌈</mo>               <mrow class="MJX-TeXAtom-ORD">                 <mfrac>                   <mi>n</mi>                   <mn>2</mn>                 </mfrac>               </mrow>               <mo>⌉</mo>             </mrow>             <mo>−<!-- − --></mo>             <mn>1</mn>           </mrow>         </munderover>         <mo stretchy="false">(</mo>         <mi>n</mi>         <mo>−<!-- − --></mo>         <mn>2</mn>         <mi>k</mi>         <mo stretchy="false">)</mo>         <mo>=</mo>         <mi>n</mi>         <mo stretchy="false">(</mo>         <mi>n</mi>         <mo>−<!-- − --></mo>         <mn>2</mn>         <mo stretchy="false">)</mo>         <mo stretchy="false">(</mo>         <mi>n</mi>         <mo>−<!-- − --></mo>         <mn>4</mn>         <mo stretchy="false">)</mo>         <mo>⋯<!-- ⋯ --></mo>         <mo>.</mo>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle n!!=\\\\prod _{k=0}^{\\\\left\\\\lceil {\\rac {n}{2}}\\
ight\\
ceil -1}(n-2k)=n(n-2)(n-4)\\\\cdots .}</annotation>   </semantics> </math></div><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd4bb531283bcca76191054e0a1dfc761bc9265e" class="mwe-math-fallback-image-display mw-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:42.729ex; height:9.009ex;" alt="{\\\\displaystyle n!!=\\\\prod _{k=0}^{\\\\left\\\\lceil {\\rac {n}{2}}\\
ight\\
ceil -1}(n-2k)=n(n-2)(n-4)\\\\cdots .}"></div> Restated, this says that for even <span class="texhtml mvar" style="font-style:italic;">n</span>, the double factorial is <div class="mwe-math-element"><div class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle n!!=\\\\prod _{k=1}^{\\rac {n}{2}}(2k)=n(n-2)(n-4)\\\\cdots 4\\\\cdot 2\\\\,,}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi>n</mi>         <mo>!</mo>         <mo>!</mo>         <mo>=</mo>         <munderover>           <mo>∏<!-- ∏ --></mo>           <mrow class="MJX-TeXAtom-ORD">             <mi>k</mi>             <mo>=</mo>             <mn>1</mn>           </mrow>           <mrow class="MJX-TeXAtom-ORD">             <mfrac>               <mi>n</mi>               <mn>2</mn>             </mfrac>           </mrow>         </munderover>         <mo stretchy="false">(</mo>         <mn>2</mn>         <mi>k</mi>         <mo stretchy="false">)</mo>         <mo>=</mo>         <mi>n</mi>         <mo stretchy="false">(</mo>         <mi>n</mi>         <mo>−<!-- − --></mo>         <mn>2</mn>         <mo stretchy="false">)</mo>         <mo stretchy="false">(</mo>         <mi>n</mi>         <mo>−<!-- − --></mo>         <mn>4</mn>         <mo stretchy="false">)</mo>         <mo>⋯<!-- ⋯ --></mo>         <mn>4</mn>         <mo>⋅<!-- ⋅ --></mo>         <mn>2</mn>         <mspace width="thinmathspace"></mspace>         <mo>,</mo>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle n!!=\\\\prod _{k=1}^{\\rac {n}{2}}(2k)=n(n-2)(n-4)\\\\cdots 4\\\\cdot 2\\\\,,}</annotation>   </semantics> </math></div><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a2c6b3ce11a2addbe78d4bdf74e3f030df3296d" class="mwe-math-fallback-image-display mw-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:40.382ex; height:8.343ex;" alt="{\\\\displaystyle n!!=\\\\prod _{k=1}^{\\rac {n}{2}}(2k)=n(n-2)(n-4)\\\\cdots 4\\\\cdot 2\\\\,,}"></div> while for odd <span class="texhtml mvar" style="font-style:italic;">n</span> it is <div class="mwe-math-element"><div class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle n!!=\\\\prod _{k=1}^{\\rac {n+1}{2}}(2k-1)=n(n-2)(n-4)\\\\cdots 3\\\\cdot 1\\\\,.}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi>n</mi>         <mo>!</mo>         <mo>!</mo>         <mo>=</mo>         <munderover>           <mo>∏<!-- ∏ --></mo>           <mrow class="MJX-TeXAtom-ORD">             <mi>k</mi>             <mo>=</mo>             <mn>1</mn>           </mrow>           <mrow class="MJX-TeXAtom-ORD">             <mfrac>               <mrow>                 <mi>n</mi>                 <mo>+</mo>                 <mn>1</mn>               </mrow>               <mn>2</mn>             </mfrac>           </mrow>         </munderover>         <mo stretchy="false">(</mo>         <mn>2</mn>         <mi>k</mi>         <mo>−<!-- − --></mo>         <mn>1</mn>         <mo stretchy="false">)</mo>         <mo>=</mo>         <mi>n</mi>         <mo stretchy="false">(</mo>         <mi>n</mi>         <mo>−<!-- − --></mo>         <mn>2</mn>         <mo stretchy="false">)</mo>         <mo stretchy="false">(</mo>         <mi>n</mi>         <mo>−<!-- − --></mo>         <mn>4</mn>         <mo stretchy="false">)</mo>         <mo>⋯<!-- ⋯ --></mo>         <mn>3</mn>         <mo>⋅<!-- ⋅ --></mo>         <mn>1</mn>         <mspace width="thinmathspace"></mspace>         <mo>.</mo>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle n!!=\\\\prod _{k=1}^{\\rac {n+1}{2}}(2k-1)=n(n-2)(n-4)\\\\cdots 3\\\\cdot 1\\\\,.}</annotation>   </semantics> </math></div><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc7e4a63cca4a1cc136e5b72bd2de4db575d8c83" class="mwe-math-fallback-image-display mw-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:44.757ex; height:8.676ex;" alt="{\\\\displaystyle n!!=\\\\prod _{k=1}^{\\rac {n+1}{2}}(2k-1)=n(n-2)(n-4)\\\\cdots 3\\\\cdot 1\\\\,.}"></div> For example, <span class="texhtml">9‼ = 9 × 7 × 5 × 3 × 1 = 945</span>. The zero double factorial <span class="texhtml">0‼ = 1</span> as an empty product 
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Double_factorial">https://en.wikipedia.org/wiki/Double_factorial</a>)"""@en ;
  skos:prefLabel "double factorielle"@fr, "double factorial"@en ;
  dc:created "2023-08-24"^^xsd:date ;
  a skos:Concept ;
  skos:exactMatch <https://fr.wikipedia.org/wiki/Analogues_de_la_factorielle#Multifactorielles>, <https://en.wikipedia.org/wiki/Double_factorial> ;
  skos:broader psr:-W127WDLN-J ;
  skos:inScheme psr: .