@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:-Q1JDMF7N-Q .

psr: a skos:ConceptScheme .
psr:-Q1JDMF7N-Q
  skos:definition """In mathematics, the <b>Euler numbers</b> are a sequence <i>E<sub>n</sub></i> of integers (sequence <span class="nowrap external">A122045</span> in the OEIS) defined by the Taylor series expansion
<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 {\\rac {1}{\\\\cosh t}}={\\rac {2}{e^{t}+e^{-t}}}=\\\\sum _{n=0}^{\\\\infty }{\\rac {E_{n}}{n!}}\\\\cdot t^{n}}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mfrac>
<br/>            <mn>1</mn>
<br/>            <mrow>
<br/>              <mi>cosh</mi>
<br/>              <mo>⁡<!-- ⁡ --></mo>
<br/>              <mi>t</mi>
<br/>            </mrow>
<br/>          </mfrac>
<br/>        </mrow>
<br/>        <mo>=</mo>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mfrac>
<br/>            <mn>2</mn>
<br/>            <mrow>
<br/>              <msup>
<br/>                <mi>e</mi>
<br/>                <mrow class="MJX-TeXAtom-ORD">
<br/>                  <mi>t</mi>
<br/>                </mrow>
<br/>              </msup>
<br/>              <mo>+</mo>
<br/>              <msup>
<br/>                <mi>e</mi>
<br/>                <mrow class="MJX-TeXAtom-ORD">
<br/>                  <mo>−<!-- − --></mo>
<br/>                  <mi>t</mi>
<br/>                </mrow>
<br/>              </msup>
<br/>            </mrow>
<br/>          </mfrac>
<br/>        </mrow>
<br/>        <mo>=</mo>
<br/>        <munderover>
<br/>          <mo>∑<!-- ∑ --></mo>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>n</mi>
<br/>            <mo>=</mo>
<br/>            <mn>0</mn>
<br/>          </mrow>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi mathvariant="normal">∞<!-- ∞ --></mi>
<br/>          </mrow>
<br/>        </munderover>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mfrac>
<br/>            <msub>
<br/>              <mi>E</mi>
<br/>              <mrow class="MJX-TeXAtom-ORD">
<br/>                <mi>n</mi>
<br/>              </mrow>
<br/>            </msub>
<br/>            <mrow>
<br/>              <mi>n</mi>
<br/>              <mo>!</mo>
<br/>            </mrow>
<br/>          </mfrac>
<br/>        </mrow>
<br/>        <mo>⋅<!-- ⋅ --></mo>
<br/>        <msup>
<br/>          <mi>t</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>n</mi>
<br/>          </mrow>
<br/>        </msup>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle {\\rac {1}{\\\\cosh t}}={\\rac {2}{e^{t}+e^{-t}}}=\\\\sum _{n=0}^{\\\\infty }{\\rac {E_{n}}{n!}}\\\\cdot t^{n}}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bef01280cc39c7321fbfc295a5babae89a6ed906" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -3.005ex; width:32.686ex; height:6.843ex;" alt="{\\\\displaystyle {\\rac {1}{\\\\cosh t}}={\\rac {2}{e^{t}+e^{-t}}}=\\\\sum _{n=0}^{\\\\infty }{\\rac {E_{n}}{n!}}\\\\cdot t^{n}}"></span>,</dd></dl>
<br/>where <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 \\\\cosh(t)}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>cosh</mi>
<br/>        <mo>⁡<!-- ⁡ --></mo>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>t</mi>
<br/>        <mo stretchy="false">)</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle \\\\cosh(t)}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/912c3e889ebd83f0883a8a0e7fe10b27b30799d3" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:7.053ex; height:2.843ex;" alt="{\\\\displaystyle \\\\cosh(t)}"></span> is the hyperbolic cosine function. The Euler numbers are related to a special value of the Euler polynomials, namely:
<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 E_{n}=2^{n}E_{n}({\\	frac {1}{2}}).}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <msub>
<br/>          <mi>E</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>n</mi>
<br/>          </mrow>
<br/>        </msub>
<br/>        <mo>=</mo>
<br/>        <msup>
<br/>          <mn>2</mn>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>n</mi>
<br/>          </mrow>
<br/>        </msup>
<br/>        <msub>
<br/>          <mi>E</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>n</mi>
<br/>          </mrow>
<br/>        </msub>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mstyle displaystyle="false" scriptlevel="0">
<br/>            <mfrac>
<br/>              <mn>1</mn>
<br/>              <mn>2</mn>
<br/>            </mfrac>
<br/>          </mstyle>
<br/>        </mrow>
<br/>        <mo stretchy="false">)</mo>
<br/>        <mo>.</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle E_{n}=2^{n}E_{n}({\\	frac {1}{2}}).}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d5c7e279225f2ec5e0947c7b2bf46dea201250f" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -1.171ex; width:15.461ex; height:3.509ex;" alt="{\\\\displaystyle E_{n}=2^{n}E_{n}({\\	frac {1}{2}}).}"></span></dd></dl>
<br/>The Euler numbers appear in the Taylor series expansions of the secant and hyperbolic secant functions. The latter is the function in the definition. They also occur in combinatorics, specifically when counting the number of alternating permutations of a set with an even number of elements. 
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Euler_numbers">https://en.wikipedia.org/wiki/Euler_numbers</a>)"""@en, """Les <b>nombres d'Euler</b> <i>E<sub>n</sub></i> forment une suite d'entiers naturels définis par le développement en série de Taylor suivant&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 {\\rac {1}{\\\\cos x}}=\\\\sum _{n=0}^{\\\\infty }E_{n}{\\rac {x^{n}}{n!}}}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mfrac>
<br/>            <mn>1</mn>
<br/>            <mrow>
<br/>              <mi>cos</mi>
<br/>              <mo>⁡<!-- ⁡ --></mo>
<br/>              <mi>x</mi>
<br/>            </mrow>
<br/>          </mfrac>
<br/>        </mrow>
<br/>        <mo>=</mo>
<br/>        <munderover>
<br/>          <mo>∑<!-- ∑ --></mo>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>n</mi>
<br/>            <mo>=</mo>
<br/>            <mn>0</mn>
<br/>          </mrow>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi mathvariant="normal">∞<!-- ∞ --></mi>
<br/>          </mrow>
<br/>        </munderover>
<br/>        <msub>
<br/>          <mi>E</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>n</mi>
<br/>          </mrow>
<br/>        </msub>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mfrac>
<br/>            <msup>
<br/>              <mi>x</mi>
<br/>              <mrow class="MJX-TeXAtom-ORD">
<br/>                <mi>n</mi>
<br/>              </mrow>
<br/>            </msup>
<br/>            <mrow>
<br/>              <mi>n</mi>
<br/>              <mo>!</mo>
<br/>            </mrow>
<br/>          </mfrac>
<br/>        </mrow>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle {\\rac {1}{\\\\cos x}}=\\\\sum _{n=0}^{\\\\infty }E_{n}{\\rac {x^{n}}{n!}}}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d7db89f3a4c1ea52fcfefac7eb3f2e7ea9534bb" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -3.005ex; width:18.823ex; height:6.843ex;" alt="{\\rac  {1}{\\\\cos x}}=\\\\sum _{{n=0}}^{{\\\\infty }}E_{n}{\\rac  {x^{n}}{n!}}"></span></dd></dl>
<br/>On les appelle aussi parfois les nombres sécants ou nombres zig-zag. 
<br/>(Wikipedia, L'Encylopédie Libre, <a href="https://fr.wikipedia.org/wiki/Nombre_d%27Euler">https://fr.wikipedia.org/wiki/Nombre_d%27Euler</a>)"""@fr ;
  dc:created "2023-07-25"^^xsd:date ;
  skos:exactMatch <https://en.wikipedia.org/wiki/Euler_numbers>, <https://fr.wikipedia.org/wiki/Nombre_d%27Euler> ;
  a skos:Concept ;
  skos:inScheme psr: ;
  skos:broader psr:-FM1M1PDT-5, psr:-B373Q2P1-V ;
  skos:prefLabel "Euler number"@en, "nombre d'Euler"@fr ;
  dc:modified "2023-07-25"^^xsd:date .

psr:-FM1M1PDT-5
  skos:prefLabel "suite d'entiers"@fr, "integer sequence"@en ;
  a skos:Concept ;
  skos:narrower psr:-Q1JDMF7N-Q .