Numeri euleriani - Wikipedia

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vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Formula_chiusa"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Formula chiusa</span> </div> </a> <ul id="toc-Formula_chiusa-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Proprietà_della_somma" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Proprietà_della_somma"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Proprietà della somma</span> </div> </a> <ul id="toc-Proprietà_della_somma-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Identità" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Identità"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Identità</span> </div> </a> <ul id="toc-Identità-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Numeri_euleriani_di_seconda_specie" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Numeri_euleriani_di_seconda_specie"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Numeri euleriani di seconda specie</span> </div> </a> <ul id="toc-Numeri_euleriani_di_seconda_specie-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bibliografia" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Bibliografia"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Bibliografia</span> </div> </a> <ul id="toc-Bibliografia-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Voci_correlate" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Voci_correlate"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Voci 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</div> </nav> <nav class="vector-appearance-landmark" aria-label="Aspetto"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Aspetto</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">sposta nella barra laterale</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">nascondi</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">Da Wikipedia, l'enciclopedia libera.</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="it" dir="ltr"><p>In <a href="/wiki/Combinatoria" title="Combinatoria">combinatoria</a>, il <b>numero euleriano</b> <i>A</i>(<i>n</i>, <i>m</i>) è il numero di <a href="/wiki/Permutazioni" class="mw-redirect" title="Permutazioni">permutazioni</a> dei numeri fra 1 e <i>n</i> nelle quali esattamente <i>m</i> elementi sono maggiori di quelli precedenti. Tali numeri sono anche i coefficienti dei <b>polinomi di Eulero</b>: </p> <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 A_{n}(t)=\sum _{m=0}^{n}A(n,m)\ t^{m}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mi>A</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi>m</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{n}(t)=\sum _{m=0}^{n}A(n,m)\ t^{m}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2482ca8ff29ff094cf42f6ea951b4b569abaabce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:24.403ex; height:6.843ex;" alt="{\displaystyle A_{n}(t)=\sum _{m=0}^{n}A(n,m)\ t^{m}.}"></span></dd></dl> <p>I polinomi di Eulero sono definiti dalla funzione generatrice esponenziale: </p> <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 \sum _{n=0}^{\infty }A_{n}(t)\,{\frac {x^{n}}{n!}}={\frac {t-1}{t-\mathrm {e} ^{(t-1)\,x}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mrow> <mi>n</mi> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>t</mi> <mo>−</mo> <mn>1</mn> </mrow> <mrow> <mi>t</mi> <mo>−</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>x</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{n=0}^{\infty }A_{n}(t)\,{\frac {x^{n}}{n!}}={\frac {t-1}{t-\mathrm {e} ^{(t-1)\,x}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de3cf5f80fbf7e750a2e03ab68546a4fb16d7eea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:27.951ex; height:6.843ex;" alt="{\displaystyle \sum _{n=0}^{\infty }A_{n}(t)\,{\frac {x^{n}}{n!}}={\frac {t-1}{t-\mathrm {e} ^{(t-1)\,x}}}.}"></span></dd></dl> <p>Essi possono essere calcolati attraverso la seguente formula ricorsiva: </p> <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 A_{0}(t)=1,\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{0}(t)=1,\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2cde59eeccd33d008206a71413c322bdc5c2bd19" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.741ex; height:2.843ex;" alt="{\displaystyle A_{0}(t)=1,\,}"></span></dd></dl> <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 A_{n}(t)=t\,(1-t)\,A_{n-1}'(t)+A_{n-1}(t)\,(1+(n-1)\,t),\quad n\geq 1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>t</mi> <mspace width="thinmathspace" /> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <msubsup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> <mo>′</mo> </msubsup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>t</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="1em" /> <mi>n</mi> <mo>≥</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{n}(t)=t\,(1-t)\,A_{n-1}'(t)+A_{n-1}(t)\,(1+(n-1)\,t),\quad n\geq 1.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/664d4663fc4f78f42329df76e38c798459771b40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:59.529ex; height:3.176ex;" alt="{\displaystyle A_{n}(t)=t\,(1-t)\,A_{n-1}'(t)+A_{n-1}(t)\,(1+(n-1)\,t),\quad n\geq 1.}"></span></dd></dl> <p>Un modo equivalente per dare questa definizione è quello di definire i polinomi di Eulero induttivamente: </p> <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 A_{0}(t)=1,\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{0}(t)=1,\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2cde59eeccd33d008206a71413c322bdc5c2bd19" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.741ex; height:2.843ex;" alt="{\displaystyle A_{0}(t)=1,\,}"></span></dd></dl> <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 A_{n}(t)=\sum _{k=0}^{n-1}{\binom {n}{k}}\,A_{k}(t)\,(t-1)^{n-1-k},\quad n\geq 1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</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"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <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> <mspace width="thinmathspace" /> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo>−</mo> <mi>k</mi> </mrow> </msup> <mo>,</mo> <mspace width="1em" /> <mi>n</mi> <mo>≥</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{n}(t)=\sum _{k=0}^{n-1}{\binom {n}{k}}\,A_{k}(t)\,(t-1)^{n-1-k},\quad n\geq 1.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3992216b9d4a3ba24388a16de5f67ae24665a9f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:45.287ex; height:7.509ex;" alt="{\displaystyle A_{n}(t)=\sum _{k=0}^{n-1}{\binom {n}{k}}\,A_{k}(t)\,(t-1)^{n-1-k},\quad n\geq 1.}"></span></dd></dl> <p>Le notazioni per questi numeri sono <i>A</i>(<i>n</i>, <i>m</i>), <i>E</i>(<i>n</i>, <i>m</i>) e <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 \scriptstyle \left\langle {n \atop m}\right\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow> <mo>⟨</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac linethickness="0"> <mi>n</mi> <mi>m</mi> </mfrac> </mrow> <mo>⟩</mo> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle \left\langle {n \atop m}\right\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a8fef16b6f90ee4b95edd0230c459daee11d516" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.281ex; height:2.509ex;" alt="{\displaystyle \scriptstyle \left\langle {n \atop m}\right\rangle }"></span>. </p><p>Essi non vanno confusi con i <a href="/wiki/Numero_di_Eulero_(teoria_dei_numeri)" class="mw-redirect" title="Numero di Eulero (teoria dei numeri)">numeri di Eulero</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Storia">Storia</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Numeri_euleriani&veaction=edit&section=1" title="Modifica la sezione Storia" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Numeri_euleriani&action=edit&section=1" title="Edit section's source code: Storia"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:EulerianPolynomialsByEuler1755.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/43/EulerianPolynomialsByEuler1755.png/220px-EulerianPolynomialsByEuler1755.png" decoding="async" width="220" height="161" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/43/EulerianPolynomialsByEuler1755.png/330px-EulerianPolynomialsByEuler1755.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/43/EulerianPolynomialsByEuler1755.png/440px-EulerianPolynomialsByEuler1755.png 2x" data-file-width="909" data-file-height="666" /></a><figcaption>Polinomi di Eulero</figcaption></figure> <p>Nel 1755 <a href="/wiki/Eulero" title="Eulero">Eulero</a> si occupò, nel libro <i>Institutiones calculi differentialis</i>, dei polinomi <i>α</i><sub>1</sub>(<i>x</i>) = 1, <i>α</i><sub>2</sub>(<i>x</i>) = <i>x</i> + 1, <i>α</i><sub>3</sub>(<i>x</i>) = <i>x</i><sup>2</sup> + 4<i>x</i> + 1, ecc. Tali polinomi sono una variante di quelli che oggi sono chiamati polinomi di Eulero <i>A</i><i><sub>n</sub></i>(<i>x</i>). </p> <div class="mw-heading mw-heading2"><h2 id="Proprietà"><span id="Propriet.C3.A0"></span>Proprietà</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Numeri_euleriani&veaction=edit&section=2" title="Modifica la sezione Proprietà" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Numeri_euleriani&action=edit&section=2" title="Edit section's source code: Proprietà"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Per ogni valore <i>n</i> > 0, l'indice <i>m</i> in <i>A</i>(<i>n</i>, <i>m</i>) può assumere valori compresi tra 0 e <i>n</i> − 1. Per <i>n </i>dato, esiste una sola permutazione con nessun valore maggiore di quello che lo precede; è la permutazione (<i>n</i>, <i>n</i> − 1, <i>n</i> − 2, ..., 1). Inoltre ne esiste una sola con <i>n</i> − 1 valori maggiori del precedente; è la permutazione (1, 2, 3, ..., <i>n</i>). Perciò, <i>A</i>(<i>n</i>, 0) e <i>A</i>(<i>n</i>, <i>n</i> − 1) valgono 1 per ogni valore di <i>n</i>. </p><p>L'inversione di una permutazione con <i>m</i> numeri maggiori dei rispettivi numeri precedenti<i> </i>genera un'altra permutazione in cui tali valori sono in quantità <i>n</i> − <i>m</i> − 1. Dunque <i>A</i>(<i>n</i>, <i>m</i>) = <i>A</i>(<i>n</i>, <i>n</i> − <i>m</i> − 1). </p><p>I valori di <i>A</i>(<i>n</i>, <i>m</i>) possono essere calcolati a mano per valori piccoli di <i>n</i> e <i>m</i>. Ad esempio, per <i>n</i> ≤ 3, si ha: </p> <dl><dd><table class="wikitable" style="margin-bottom: 10px;"> <tbody><tr> <th><i>n</i> </th> <th><i>m</i> </th> <th>Permutazioni </th> <th><i>A</i>(<i>n</i>, <i>m</i>) </th></tr> <tr> <td>1 </td> <td>0 </td> <td>(1) </td> <td><i>A</i>(1,0) = 1 </td></tr> <tr> <td rowspan="2">2 </td> <td>0 </td> <td>(2, 1) </td> <td><i>A</i>(2,0) = 1 </td></tr> <tr> <td>1 </td> <td>(1, <b>2</b>) </td> <td><i>A</i>(2,1) = 1 </td></tr> <tr> <td rowspan="3">3 </td> <td>0 </td> <td>(3, 2, 1) </td> <td><i>A</i>(3,0) = 1 </td></tr> <tr> <td>1 </td> <td>(1, <b>3</b>, 2) (2, 1, <b>3</b>) (2, <b>3</b>, 1) (3, 1, <b>2</b>) </td> <td><i>A</i>(3,1) = 4 </td></tr> <tr> <td>2 </td> <td>(1, <b>2</b>, <b>3</b>) </td> <td><i>A</i>(3,2) = 1 </td></tr></tbody></table></dd></dl> <p>Per valori più grandi di <i>n</i>, <i>A</i>(<i>n</i>, <i>m</i>) si può calcolare usando la ricorsione </p> <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 A(n,m)=(n-m)A(n-1,m-1)+(m+1)A(n-1,m).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi>m</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mi>m</mi> <mo stretchy="false">)</mo> <mi>A</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mi>m</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mi>m</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>A</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mi>m</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A(n,m)=(n-m)A(n-1,m-1)+(m+1)A(n-1,m).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ccdd4fe49eee8af9641280cd4cbd5342381f8a06" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:58.596ex; height:2.843ex;" alt="{\displaystyle A(n,m)=(n-m)A(n-1,m-1)+(m+1)A(n-1,m).}"></span></dd></dl> <p>Da cui, ad esempio: </p> <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 A(4,1)=(4-1)A(3,0)+(1+1)A(3,1)=3\times 1+2\times 4=11.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo stretchy="false">(</mo> <mn>4</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mn>4</mn> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>A</mi> <mo stretchy="false">(</mo> <mn>3</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>A</mi> <mo stretchy="false">(</mo> <mn>3</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>3</mn> <mo>×</mo> <mn>1</mn> <mo>+</mo> <mn>2</mn> <mo>×</mo> <mn>4</mn> <mo>=</mo> <mn>11.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A(4,1)=(4-1)A(3,0)+(1+1)A(3,1)=3\times 1+2\times 4=11.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/228458c2f59f5c1dc8ff853074a21b9984e65da1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:62.962ex; height:2.843ex;" alt="{\displaystyle A(4,1)=(4-1)A(3,0)+(1+1)A(3,1)=3\times 1+2\times 4=11.}"></span></dd></dl> <p>I valori di <i>A</i>(<i>n</i>, <i>m</i>) (<span class="nowrap">sequenza <a href="//oeis.org/A008292" class="extiw" title="oeis:A008292">A008292</a></span><span> dell'</span><a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a><span>)</span> per 0 ≤ <i>n</i> ≤ 9 sono: </p> <dl><dd><table class="wikitable" style="margin-bottom: 10px;"> <tbody><tr> <th><i>n</i> \ <i>m</i> </th> <th width="50">0 </th> <th width="50">1 </th> <th width="50">2 </th> <th width="50">3 </th> <th width="50">4 </th> <th width="50">5 </th> <th width="50">6 </th> <th width="50">7 </th> <th width="50">8 </th></tr> <tr> <th>1 </th> <td>1 </td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td></tr> <tr> <th>2 </th> <td>1 </td> <td>1 </td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td></tr> <tr> <th>3 </th> <td>1 </td> <td>4 </td> <td>1 </td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td></tr> <tr> <th>4 </th> <td>1 </td> <td>11 </td> <td>11 </td> <td>1 </td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td></tr> <tr> <th>5 </th> <td>1 </td> <td>26 </td> <td>66 </td> <td>26 </td> <td>1 </td> <td> </td> <td> </td> <td> </td> <td> </td></tr> <tr> <th>6 </th> <td>1 </td> <td>57 </td> <td>302 </td> <td>302 </td> <td>57 </td> <td>1 </td> <td> </td> <td> </td> <td> </td></tr> <tr> <th>7 </th> <td>1 </td> <td>120 </td> <td>1191 </td> <td>2416 </td> <td>1191 </td> <td>120 </td> <td>1 </td> <td> </td> <td> </td></tr> <tr> <th>8 </th> <td>1 </td> <td>247 </td> <td>4293 </td> <td>15619 </td> <td>15619 </td> <td>4293 </td> <td>247 </td> <td>1 </td> <td> </td></tr> <tr> <th>9 </th> <td>1 </td> <td>502 </td> <td>14608 </td> <td>88234 </td> <td>156190 </td> <td>88234 </td> <td>14608 </td> <td>502 </td> <td>1 </td></tr></tbody></table></dd></dl> <p>Questa disposizione triangolare si chiama <b>triangolo di Eulero</b> e condivide alcune caratteristiche con il <a href="/wiki/Triangolo_di_Tartaglia" title="Triangolo di Tartaglia">triangolo di Tartaglia</a>. La somma dei numeri sulla riga <i>n-esima</i> è <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> <mo>!</mo> </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/bae971720be3cc9b8d82f4cdac89cb89877514a6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.042ex; height:2.176ex;" alt="{\displaystyle n!}"></span>. </p> <div class="mw-heading mw-heading2"><h2 id="Formula_chiusa">Formula chiusa</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Numeri_euleriani&veaction=edit&section=3" title="Modifica la sezione Formula chiusa" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Numeri_euleriani&action=edit&section=3" title="Edit section's source code: Formula chiusa"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Una <a href="/wiki/Forma_chiusa" class="mw-disambig" title="Forma chiusa">forma chiusa</a> per <i>A</i>(<i>n</i>, <i>m</i>) è la seguente: </p> <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 A(n,m)=\sum _{k=0}^{m+1}(-1)^{k}{\binom {n+1}{k}}(m+1-k)^{n}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi>m</mi> <mo stretchy="false">)</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"> <mi>m</mi> <mo>+</mo> <mn>1</mn> </mrow> </munderover> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> <mi>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>m</mi> <mo>+</mo> <mn>1</mn> <mo>−</mo> <mi>k</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A(n,m)=\sum _{k=0}^{m+1}(-1)^{k}{\binom {n+1}{k}}(m+1-k)^{n}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0fc3fe587d72cbc8006949bf3edad0cfad3d420c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:43.12ex; height:7.509ex;" alt="{\displaystyle A(n,m)=\sum _{k=0}^{m+1}(-1)^{k}{\binom {n+1}{k}}(m+1-k)^{n}.}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Proprietà_della_somma"><span id="Propriet.C3.A0_della_somma"></span>Proprietà della somma</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Numeri_euleriani&veaction=edit&section=4" title="Modifica la sezione Proprietà della somma" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Numeri_euleriani&action=edit&section=4" title="Edit section's source code: Proprietà della somma"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>È evidente dalla definizione di combinatoria che la somma dei numeri di Eulero per un dato valore di <i>n</i> è il numero totale di permutazioni dei numeri tra 1 e <i>n</i>, ovvero </p> <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 \sum _{m=0}^{n-1}A(n,m)=n!{\text{ per }}n\geq 1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <mi>A</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi>m</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>n</mi> <mo>!</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext> per </mtext> </mrow> <mi>n</mi> <mo>≥</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{m=0}^{n-1}A(n,m)=n!{\text{ per }}n\geq 1.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af6b0a33515bbac0a9f1a57b27684a1c622442ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:27.792ex; height:7.343ex;" alt="{\displaystyle \sum _{m=0}^{n-1}A(n,m)=n!{\text{ per }}n\geq 1.}"></span></dd></dl> <p>La <a href="/wiki/Serie_alternata" title="Serie alternata">serie alternata</a> dei numeri di Eulero per <i>n</i> dato è strettamente collegata ai <a href="/wiki/Numeri_di_Bernoulli" title="Numeri di Bernoulli">numeri di Bernoulli</a> <i>B</i><sub><i>n</i>+1</sub> </p> <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 \sum _{m=0}^{n-1}(-1)^{m}A(n,m)={\frac {2^{n+1}(2^{n+1}-1)B_{n+1}}{n+1}}{\text{ per }}n\geq 1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mi>A</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi>m</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mrow> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext> per </mtext> </mrow> <mi>n</mi> <mo>≥</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{m=0}^{n-1}(-1)^{m}A(n,m)={\frac {2^{n+1}(2^{n+1}-1)B_{n+1}}{n+1}}{\text{ per }}n\geq 1.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/399b0b8b1362c60a0ca6c65a4f72a7b7aff661ec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:52.513ex; height:7.343ex;" alt="{\displaystyle \sum _{m=0}^{n-1}(-1)^{m}A(n,m)={\frac {2^{n+1}(2^{n+1}-1)B_{n+1}}{n+1}}{\text{ per }}n\geq 1.}"></span></dd></dl> <p>Altre sommatorie interessanti per i numeri di Eulero sono: </p> <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 \sum _{m=0}^{n-1}(-1)^{m}{\frac {A(n,m)}{\binom {n-1}{m}}}=0{\text{ per }}n\geq 2,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>A</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi>m</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> <mi>m</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext> per </mtext> </mrow> <mi>n</mi> <mo>≥</mo> <mn>2</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{m=0}^{n-1}(-1)^{m}{\frac {A(n,m)}{\binom {n-1}{m}}}=0{\text{ per }}n\geq 2,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0203918048fdfa0bcbdb883d458536086102ae42" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:33.817ex; height:7.509ex;" alt="{\displaystyle \sum _{m=0}^{n-1}(-1)^{m}{\frac {A(n,m)}{\binom {n-1}{m}}}=0{\text{ per }}n\geq 2,}"></span></dd></dl> <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 \sum _{m=0}^{n-1}(-1)^{m}{\frac {A(n,m)}{\binom {n}{m}}}=(n+1)B_{n}{\text{ per }}n\geq 2,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>A</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi>m</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mi>m</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mtext> per </mtext> </mrow> <mi>n</mi> <mo>≥</mo> <mn>2</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{m=0}^{n-1}(-1)^{m}{\frac {A(n,m)}{\binom {n}{m}}}=(n+1)B_{n}{\text{ per }}n\geq 2,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b038b12a8d05ee782e51775de44e92356e86aeaa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:42.844ex; height:7.509ex;" alt="{\displaystyle \sum _{m=0}^{n-1}(-1)^{m}{\frac {A(n,m)}{\binom {n}{m}}}=(n+1)B_{n}{\text{ per }}n\geq 2,}"></span></dd></dl> <p>dove <i>B</i><sub><i>n</i></sub> è l'<i>n</i>-esimo numero di Bernoulli. </p> <div class="mw-heading mw-heading2"><h2 id="Identità"><span id="Identit.C3.A0"></span>Identità</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Numeri_euleriani&veaction=edit&section=5" title="Modifica la sezione Identità" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Numeri_euleriani&action=edit&section=5" title="Edit section's source code: Identità"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>I numeri di Eulero compaiono nella <a href="/wiki/Funzione_generatrice" title="Funzione generatrice">funzione generatrice</a> delle sequenze di potenze <i>n</i>-esime </p> <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 \sum _{k=0}^{\infty }k^{n}x^{k}={\frac {\sum _{m=0}^{n-1}A(n,m)x^{m+1}}{(1-x)^{n+1}}}{\text{ per }}n\geq 0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <mi>A</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi>m</mi> <mo stretchy="false">)</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>x</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext> per </mtext> </mrow> <mi>n</mi> <mo>≥</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{k=0}^{\infty }k^{n}x^{k}={\frac {\sum _{m=0}^{n-1}A(n,m)x^{m+1}}{(1-x)^{n+1}}}{\text{ per }}n\geq 0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f2586b32ac2e018d24a46bdf955efeed9a2b75a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:42.968ex; height:7.509ex;" alt="{\displaystyle \sum _{k=0}^{\infty }k^{n}x^{k}={\frac {\sum _{m=0}^{n-1}A(n,m)x^{m+1}}{(1-x)^{n+1}}}{\text{ per }}n\geq 0.}"></span></dd></dl> <p>Questo implica che 0<sup>0</sup> = 0 e <i>A</i>(0,0) = 1 (poiché esiste una permutazione di 0 elementi, e nessuno di essi può essere maggiore di un altro). </p><p><b>L'identità di Worpitzky</b> permette di esprimere <i>x</i><i><sup>n</sup></i> come <a href="/wiki/Combinazione_lineare" title="Combinazione lineare">combinazione lineare</a> di numeri di Eulero con i <a href="/wiki/Coefficiente_binomiale" title="Coefficiente binomiale">coefficienti binomiali</a>: </p> <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 x^{n}=\sum _{m=0}^{n-1}A(n,m){\binom {x+m}{n}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <mi>A</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi>m</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mi>x</mi> <mo>+</mo> <mi>m</mi> </mrow> <mi>n</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{n}=\sum _{m=0}^{n-1}A(n,m){\binom {x+m}{n}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/461ce6f7dc10f96150e5aae82618eec003224191" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:27.877ex; height:7.343ex;" alt="{\displaystyle x^{n}=\sum _{m=0}^{n-1}A(n,m){\binom {x+m}{n}}.}"></span></dd></dl> <p>Da questa identità segue che </p> <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 \sum _{k=1}^{N}k^{n}=\sum _{m=0}^{n-1}A(n,m){\binom {N+1+m}{n+1}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</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> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <mi>A</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi>m</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mi>N</mi> <mo>+</mo> <mn>1</mn> <mo>+</mo> <mi>m</mi> </mrow> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{k=1}^{N}k^{n}=\sum _{m=0}^{n-1}A(n,m){\binom {N+1+m}{n+1}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b86d4d06eebabee16f18418d1cda4a911f65f38" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:36.237ex; height:7.343ex;" alt="{\displaystyle \sum _{k=1}^{N}k^{n}=\sum _{m=0}^{n-1}A(n,m){\binom {N+1+m}{n+1}}.}"></span></dd></dl> <p>Un'altra curiosa identità è </p> <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 {\frac {e}{1-ex}}=\sum _{n=0}^{\infty }{\frac {A_{n}(x)}{n!(1-x)^{n+1}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>e</mi> <mrow> <mn>1</mn> <mo>−</mo> <mi>e</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>n</mi> <mo>!</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>x</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {e}{1-ex}}=\sum _{n=0}^{\infty }{\frac {A_{n}(x)}{n!(1-x)^{n+1}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a661f45bca7320fd47bbe4acd5b6c10614512bc7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:28.078ex; height:6.843ex;" alt="{\displaystyle {\frac {e}{1-ex}}=\sum _{n=0}^{\infty }{\frac {A_{n}(x)}{n!(1-x)^{n+1}}}.}"></span></dd></dl> <p>Dove il <a href="/wiki/Numeratore" title="Numeratore">numeratore</a> delle frazioni di destra è un <a href="/wiki/Polinomio" title="Polinomio">polinomio</a> di Eulero. </p> <div class="mw-heading mw-heading2"><h2 id="Numeri_euleriani_di_seconda_specie">Numeri euleriani di seconda specie</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Numeri_euleriani&veaction=edit&section=6" title="Modifica la sezione Numeri euleriani di seconda specie" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Numeri_euleriani&action=edit&section=6" title="Edit section's source code: Numeri euleriani di seconda specie"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Le permutazioni del <a href="/wiki/Multiinsieme" title="Multiinsieme">multiinsieme</a> {1, 1, 2, 2, ···, <i>n</i>, <i>n</i>} con la proprietà che, per ogni <i>k</i>, tutti i numeri compresi tra le due occorrenze di <i>k</i> nella permutazione sono maggiori di <i>k,</i> possono essere contate attraverso il <a href="/wiki/Semifattoriale" class="mw-redirect" title="Semifattoriale">semifattoriale</a> <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-1)!!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>!</mo> <mo>!</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (2n-1)!!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ff129b00743f4df75457151ab524ae31fab2503" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.663ex; height:2.843ex;" alt="{\displaystyle (2n-1)!!}"></span>. I numeri euleriani di seconda specie, indicati con <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 \scriptstyle \left\langle \!\!\left\langle {n \atop m}\right\rangle \!\!\right\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow> <mo>⟨</mo> <mrow> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mrow> <mo>⟨</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac linethickness="0"> <mi>n</mi> <mi>m</mi> </mfrac> </mrow> <mo>⟩</mo> </mrow> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> </mrow> <mo>⟩</mo> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle \left\langle \!\!\left\langle {n \atop m}\right\rangle \!\!\right\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/809d9dfdf7e06bd8ddc4abea49af74edbb627998" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.284ex; height:2.509ex;" alt="{\displaystyle \scriptstyle \left\langle \!\!\left\langle {n \atop m}\right\rangle \!\!\right\rangle }"></span>, servono a contare il numero di permutazioni con esattamente <i>m</i> elementi che sono più grandi dell'elemento che li precede. Ad esempio, se <i>n</i> = 3 ci sono 15 permutazioni di questo tipo: </p> <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 332211,\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>332211</mn> <mo>,</mo> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 332211,\;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d1f0cd7602922136959a1f2e159f7c288f36936" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.267ex; height:2.509ex;" alt="{\displaystyle 332211,\;}"></span></dd> <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 221133,\;221331,\;223311,\;233211,\;113322,\;133221,\;331122,\;331221,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>221133</mn> <mo>,</mo> <mspace width="thickmathspace" /> <mn>221331</mn> <mo>,</mo> <mspace width="thickmathspace" /> <mn>223311</mn> <mo>,</mo> <mspace width="thickmathspace" /> <mn>233211</mn> <mo>,</mo> <mspace width="thickmathspace" /> <mn>113322</mn> <mo>,</mo> <mspace width="thickmathspace" /> <mn>133221</mn> <mo>,</mo> <mspace width="thickmathspace" /> <mn>331122</mn> <mo>,</mo> <mspace width="thickmathspace" /> <mn>331221</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 221133,\;221331,\;223311,\;233211,\;113322,\;133221,\;331122,\;331221,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6bce859f7355df44ad234df5dac831bed3a6d6d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:68.198ex; height:2.509ex;" alt="{\displaystyle 221133,\;221331,\;223311,\;233211,\;113322,\;133221,\;331122,\;331221,}"></span></dd> <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 112233,\;122133,\;112332,\;123321,\;133122,\;122331.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>112233</mn> <mo>,</mo> <mspace width="thickmathspace" /> <mn>122133</mn> <mo>,</mo> <mspace width="thickmathspace" /> <mn>112332</mn> <mo>,</mo> <mspace width="thickmathspace" /> <mn>123321</mn> <mo>,</mo> <mspace width="thickmathspace" /> <mn>133122</mn> <mo>,</mo> <mspace width="thickmathspace" /> <mn>122331.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 112233,\;122133,\;112332,\;123321,\;133122,\;122331.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9dc4e229d59fc3dfd4e33f4a8f66a9208f30f19d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:50.891ex; height:2.509ex;" alt="{\displaystyle 112233,\;122133,\;112332,\;123321,\;133122,\;122331.}"></span></dd></dl> <p>I numeri euleriani di seconda specie soddisfano la <a href="/wiki/Relazione_di_ricorrenza" title="Relazione di ricorrenza">relazione di ricorrenza</a>, che discende dalla definizione: </p> <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 \left\langle \!\!\left\langle {n \atop m}\right\rangle \!\!\right\rangle =(2n-m-1)\left\langle \!\!\left\langle {n-1 \atop m-1}\right\rangle \!\!\right\rangle +(m+1)\left\langle \!\!\left\langle {n-1 \atop m}\right\rangle \!\!\right\rangle ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>⟨</mo> <mrow> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mrow> <mo>⟨</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac linethickness="0"> <mi>n</mi> <mi>m</mi> </mfrac> </mrow> <mo>⟩</mo> </mrow> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> </mrow> <mo>⟩</mo> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo>−</mo> <mi>m</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mrow> <mo>⟨</mo> <mrow> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mrow> <mo>⟨</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac linethickness="0"> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> <mrow> <mi>m</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>⟩</mo> </mrow> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> </mrow> <mo>⟩</mo> </mrow> <mo>+</mo> <mo stretchy="false">(</mo> <mi>m</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mrow> <mo>⟨</mo> <mrow> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mrow> <mo>⟨</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac linethickness="0"> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> <mi>m</mi> </mfrac> </mrow> <mo>⟩</mo> </mrow> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> </mrow> <mo>⟩</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\langle \!\!\left\langle {n \atop m}\right\rangle \!\!\right\rangle =(2n-m-1)\left\langle \!\!\left\langle {n-1 \atop m-1}\right\rangle \!\!\right\rangle +(m+1)\left\langle \!\!\left\langle {n-1 \atop m}\right\rangle \!\!\right\rangle ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d4dcdba606dc59a5fd768a8ac8d7269150e8dc62" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:58.984ex; height:6.176ex;" alt="{\displaystyle \left\langle \!\!\left\langle {n \atop m}\right\rangle \!\!\right\rangle =(2n-m-1)\left\langle \!\!\left\langle {n-1 \atop m-1}\right\rangle \!\!\right\rangle +(m+1)\left\langle \!\!\left\langle {n-1 \atop m}\right\rangle \!\!\right\rangle ,}"></span></dd></dl> <p>con la condizione iniziale che <i>n</i> = 0, espressa attraverso le <a href="/wiki/Parentesi_di_Iverson" title="Parentesi di Iverson">parentesi di Iverson</a>: </p> <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 \left\langle \!\!\left\langle {0 \atop m}\right\rangle \!\!\right\rangle =[m=0].}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>⟨</mo> <mrow> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mrow> <mo>⟨</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac linethickness="0"> <mn>0</mn> <mi>m</mi> </mfrac> </mrow> <mo>⟩</mo> </mrow> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> </mrow> <mo>⟩</mo> </mrow> <mo>=</mo> <mo stretchy="false">[</mo> <mi>m</mi> <mo>=</mo> <mn>0</mn> <mo stretchy="false">]</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\langle \!\!\left\langle {0 \atop m}\right\rangle \!\!\right\rangle =[m=0].}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/17d4a15e0d4c20dc4e3dc8ae86c279fe6383e6b4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:19.362ex; height:6.176ex;" alt="{\displaystyle \left\langle \!\!\left\langle {0 \atop m}\right\rangle \!\!\right\rangle =[m=0].}"></span></dd></dl> <p>Analogamente, i polinomi euleriani di seconda specie, qui indicati con <i>P</i><i><sub>n</sub></i> (non esiste una notazione standard) sono </p> <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 P_{n}(x):=\sum _{m=0}^{n}\left\langle \!\!\left\langle {n \atop m}\right\rangle \!\!\right\rangle x^{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>:=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow> <mo>⟨</mo> <mrow> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mrow> <mo>⟨</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac linethickness="0"> <mi>n</mi> <mi>m</mi> </mfrac> </mrow> <mo>⟩</mo> </mrow> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> </mrow> <mo>⟩</mo> </mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{n}(x):=\sum _{m=0}^{n}\left\langle \!\!\left\langle {n \atop m}\right\rangle \!\!\right\rangle x^{m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2de01f72980c037157679beb62cf8c42e29f1ee4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:23.648ex; height:6.843ex;" alt="{\displaystyle P_{n}(x):=\sum _{m=0}^{n}\left\langle \!\!\left\langle {n \atop m}\right\rangle \!\!\right\rangle x^{m}}"></span></dd></dl> <p>e per essi vale la seguente relazione di ricorrenza: </p> <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 P_{n+1}(x)=(2nx+1)P_{n}(x)-x(x-1)P_{n}^{\prime }(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mi>x</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>x</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <msubsup> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{n+1}(x)=(2nx+1)P_{n}(x)-x(x-1)P_{n}^{\prime }(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c7ec815e20b566d723d4d8b0e6fe38fb01c3d73a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:43.759ex; height:2.843ex;" alt="{\displaystyle P_{n+1}(x)=(2nx+1)P_{n}(x)-x(x-1)P_{n}^{\prime }(x)}"></span></dd></dl> <p>con condizione iniziale </p> <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 P_{0}(x)=1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{0}(x)=1.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c829c274abab10d6e73d282e800f8b20d04ebf5f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.593ex; height:2.843ex;" alt="{\displaystyle P_{0}(x)=1.}"></span></dd></dl> <p>Quest'ultima ricorrenza può essere scritta in modo più compatto attraverso un <a href="/wiki/Fattore_di_integrazione" title="Fattore di integrazione">fattore di integrazione</a>: </p> <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 (x-1)^{-2n-2}P_{n+1}(x)=\left(x(1-x)^{-2n-1}P_{n}(x)\right)^{\prime }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2</mn> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msup> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>x</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2</mn> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x-1)^{-2n-2}P_{n+1}(x)=\left(x(1-x)^{-2n-1}P_{n}(x)\right)^{\prime }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f594a526e10c2a9c357333470a2d8ccaaf1cc84d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:46.165ex; height:3.676ex;" alt="{\displaystyle (x-1)^{-2n-2}P_{n+1}(x)=\left(x(1-x)^{-2n-1}P_{n}(x)\right)^{\prime }}"></span></dd></dl> <p>tale che la funzione razionale </p> <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 u_{n}(x):=(x-1)^{-2n}P_{n}(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>:=</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2</mn> <mi>n</mi> </mrow> </msup> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u_{n}(x):=(x-1)^{-2n}P_{n}(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8b218385d7f20220ec0221e60ae63af08d16490" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.743ex; height:3.176ex;" alt="{\displaystyle u_{n}(x):=(x-1)^{-2n}P_{n}(x)}"></span></dd></dl> <p>soddisfa la seguente relazione di ricorrenza: </p> <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 u_{n+1}=\left({\frac {x}{1-x}}u_{n}\right)^{\prime },\quad u_{0}=1,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mrow> <mn>1</mn> <mo>−</mo> <mi>x</mi> </mrow> </mfrac> </mrow> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mo>,</mo> <mspace width="1em" /> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u_{n+1}=\left({\frac {x}{1-x}}u_{n}\right)^{\prime },\quad u_{0}=1,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5342756bd2d18407104e155cf7577aaf904fbbed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:31.218ex; height:6.343ex;" alt="{\displaystyle u_{n+1}=\left({\frac {x}{1-x}}u_{n}\right)^{\prime },\quad u_{0}=1,}"></span></dd></dl> <p>da cui si ottengono i polinomi di Eulero nella forma <i>P</i><sub><i>n</i></sub>(<i>x</i>) = (1−<i>x</i>)<sup>2<i>n</i></sup> <i>u</i><sub><i>n</i></sub>(<i>x</i>), e i numeri euleriani di seconda specie come coefficienti. </p><p>Questi sono i primi valori per i numeri euleriani di seconda specie (<span class="nowrap">sequenza <a href="//oeis.org/A008517" class="extiw" title="oeis:A008517">A008517</a></span><span> dell'</span><a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>): </p> <dl><dd><table class="wikitable"> <tbody><tr> <th><i>n</i> \ <i>m</i> </th> <th width="50">0 </th> <th width="50">1 </th> <th width="50">2 </th> <th width="50">3 </th> <th width="50">4 </th> <th width="50">5 </th> <th width="50">6 </th> <th width="50">7 </th> <th width="50">8 </th></tr> <tr> <th>1 </th> <td>1 </td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td></tr> <tr> <th>2 </th> <td>1 </td> <td>2 </td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td></tr> <tr> <th>3 </th> <td>1 </td> <td>8 </td> <td>6 </td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td></tr> <tr> <th>4 </th> <td>1 </td> <td>22 </td> <td>58 </td> <td>24 </td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td></tr> <tr> <th>5 </th> <td>1 </td> <td>52 </td> <td>328 </td> <td>444 </td> <td>120 </td> <td> </td> <td> </td> <td> </td> <td> </td></tr> <tr> <th>6 </th> <td>1 </td> <td>114 </td> <td>1452 </td> <td>4400 </td> <td>3708 </td> <td>720 </td> <td> </td> <td> </td> <td> </td></tr> <tr> <th>7 </th> <td>1 </td> <td>240 </td> <td>5610 </td> <td>32120 </td> <td>58140 </td> <td>33984 </td> <td>5040 </td> <td> </td> <td> </td></tr> <tr> <th>8 </th> <td>1 </td> <td>494 </td> <td>19950 </td> <td>195800 </td> <td>644020 </td> <td>785304 </td> <td>341136 </td> <td>40320 </td> <td> </td></tr> <tr> <th>9 </th> <td>1 </td> <td>1004 </td> <td>67260 </td> <td>1062500 </td> <td>5765500 </td> <td>12440064 </td> <td>11026296 </td> <td>3733920 </td> <td>362880 </td></tr></tbody></table></dd></dl> <p>In cui, di conseguenza, la somma della riga <i>n</i>-esima (che corrisponde anche al valore di <i>P</i><i><sub>n</sub></i>(1)), è <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-1)!!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>!</mo> <mo>!</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (2n-1)!!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ff129b00743f4df75457151ab524ae31fab2503" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.663ex; height:2.843ex;" alt="{\displaystyle (2n-1)!!}"></span>. </p> <div class="mw-heading mw-heading2"><h2 id="Bibliografia">Bibliografia</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Numeri_euleriani&veaction=edit&section=7" title="Modifica la sezione Bibliografia" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Numeri_euleriani&action=edit&section=7" title="Edit section's source code: Bibliografia"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><cite class="citation libro" style="font-style:normal">(<span style="font-weight:bolder; font-size:80%"><abbr title="latino">LA</abbr></span>) Leonardo Eulero, <span style="font-style:italic;">Institutiones calculi differentialis cum eius usu in analysi finitorum ac doctrina serierum</span> [<span style="font-style:italic;">Fondamenti del calcolo differenziale, con applicazioni in analisi finita e serie</span>], Academia imperialis scientiarum Petropolitana; Berolini: Officina Michaelis, 1755.</cite></li> <li><cite class="citation libro" style="font-style:normal">(<span style="font-weight:bolder; font-size:80%"><abbr title="inglese">EN</abbr></span>) Graham, Knuth e Patashnik, <span style="font-style:italic;"><a href="/wiki/Concrete_Mathematics" title="Concrete Mathematics">Concrete Mathematics</a>: A Foundation for Computer Science</span>, 2ª ed., Addison-Wesley, 1994, pp. <a rel="nofollow" class="external text" href="https://archive.org/details/concretemathemat00grah_505/page/n280">267</a>-272.</cite></li> <li><cite class="citation pubblicazione" style="font-style:normal">(<span style="font-weight:bolder; font-size:80%"><abbr title="inglese">EN</abbr></span>) <a rel="nofollow" class="external text" href="http://resolver.sub.uni-goettingen.de/purl?GDZPPN002610515"><span style="font-style:italic;">Eulerian numbers with fractional order parameters</span></a>, in <span style="font-style:italic;">Aequationes Mathematicae</span>, vol. 46, 1993, pp. 119–142, <a href="/wiki/Digital_object_identifier" title="Digital object identifier">DOI</a>:<a rel="nofollow" class="external text" href="https://dx.doi.org/10.1007%2Fbf01834003">10.1007/bf01834003</a>.</cite></li> <li><cite class="citation libro" style="font-style:normal">(<span style="font-weight:bolder; font-size:80%"><abbr title="inglese">EN</abbr></span>) T. K. Petersen, <a rel="nofollow" class="external text" href="https://www.springer.com/us/book/9781493930906"><span style="font-style:italic;">Eulerian Numbers</span></a>, Birkhaüser, 2015.</cite></li></ul> <div class="mw-heading mw-heading2"><h2 id="Voci_correlate">Voci correlate</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Numeri_euleriani&veaction=edit&section=8" title="Modifica la sezione Voci correlate" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Numeri_euleriani&action=edit&section=8" title="Edit section's source code: Voci correlate"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Numeri_di_Eulero" title="Numeri di Eulero">Numeri di Eulero</a></li> <li><a href="/wiki/Coefficiente_binomiale" title="Coefficiente binomiale">Coefficiente binomiale</a></li> <li><a href="/wiki/Fattoriale" title="Fattoriale">Fattoriale</a></li> <li><a href="/wiki/Numeri_di_Bernoulli" title="Numeri di Bernoulli">Numeri di Bernoulli</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Collegamenti_esterni">Collegamenti esterni</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Numeri_euleriani&veaction=edit&section=9" title="Modifica la sezione Collegamenti esterni" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Numeri_euleriani&action=edit&section=9" title="Edit section's source code: Collegamenti esterni"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><cite class="citation web" style="font-style:normal">(<span style="font-weight:bolder; font-size:80%"><abbr title="inglese">EN</abbr></span>) <a rel="nofollow" class="external text" href="http://go.helms-net.de/math/binomial_new/01_12_Eulermatrix.pdf"><span style="font-style:italic;">Euler-matrix</span></a> (<span style="font-weight: bolder; font-size:80%"><abbr title="documento in formato PDF">PDF</abbr></span>), su <span style="font-style:italic;">go.helms-net.de</span>. <small>URL consultato il 7 gennaio 2017</small>.</cite></li></ul> <div class="noprint" style="width:100%; padding: 3px 0; display: flex; flex-wrap: wrap; row-gap: 4px; column-gap: 8px; box-sizing: border-box;"><div style="flex-grow: 1"><style data-mw-deduplicate="TemplateStyles:r140555418">.mw-parser-output .itwiki-template-occhiello{width:100%;line-height:25px;border:1px solid #CCF;background-color:#F0EEFF;box-sizing:border-box}.mw-parser-output .itwiki-template-occhiello-progetto{background-color:#FAFAFA}@media screen{html.skin-theme-clientpref-night .mw-parser-output .itwiki-template-occhiello{background-color:#202122;border-color:#54595D}html.skin-theme-clientpref-night .mw-parser-output .itwiki-template-occhiello-progetto{background-color:#282929}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .itwiki-template-occhiello{background-color:#202122;border-color:#54595D}html.skin-theme-clientpref-os .mw-parser-output .itwiki-template-occhiello-progetto{background-color:#282929}}</style><div class="itwiki-template-occhiello"><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Crystal128-kmplot.svg" class="mw-file-description" title="Matematica"><img alt=" " src="//upload.wikimedia.org/wikipedia/commons/thumb/a/af/Crystal128-kmplot.svg/25px-Crystal128-kmplot.svg.png" decoding="async" width="25" height="25" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/af/Crystal128-kmplot.svg/38px-Crystal128-kmplot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/af/Crystal128-kmplot.svg/50px-Crystal128-kmplot.svg.png 2x" data-file-width="245" data-file-height="244" /></a></span> <b><a href="/wiki/Portale:Matematica" title="Portale:Matematica">Portale Matematica</a></b>: accedi alle voci di Wikipedia che trattano di Matematica</div></div></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; 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