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فاکتوریل دوبل - ویکیپدیا، دانشنامهٔ آزاد
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id="bodyContent" class="content"> <div id="mw-content-text" class="mw-body-content"> <script>function mfTempOpenSection(id){var block=document.getElementById("mf-section-"+id);block.className+=" open-block";block.previousSibling.className+=" open-block";}</script> <div class="mw-content-rtl mw-parser-output" lang="fa" dir="rtl"> <section class="mf-section-0" id="mf-section-0"> <p><b>فاکتوریل دوبل</b> یک عملگر ریاضی شبیه عمل <a href="https://fa-m-wikipedia-org.translate.goog/wiki/%D9%81%D8%A7%DA%A9%D8%AA%D9%88%D8%B1%DB%8C%D9%84?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="فاکتوریل">فاکتوریل</a> است و با نماد <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 !!}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo> ! </mo> <mo> ! </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle !!} </annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a6e2480ece878ba9a96d09f1fe710c7117f82f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.294ex; height:2.009ex;" alt="{\displaystyle !!}"></span> نمایش داده میشود.</p> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(1)"> <span class="indicator mf-icon mf-icon-expand mf-icon--small"></span> <h2 id="تعریف_عملگر"><span id=".D8.AA.D8.B9.D8.B1.DB.8C.D9.81_.D8.B9.D9.85.D9.84.DA.AF.D8.B1"></span>تعریف عملگر</h2><span class="mw-editsection"> <a role="button" href="https://fa-m-wikipedia-org.translate.goog/w/index.php?title=%D9%81%D8%A7%DA%A9%D8%AA%D9%88%D8%B1%DB%8C%D9%84_%D8%AF%D9%88%D8%A8%D9%84&action=edit&section=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="ویرایش بخش: تعریف عملگر" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>ویرایش</span> </a> </span> </div> <section class="mf-section-1 collapsible-block" id="mf-section-1"> <p>فاکتوریل دوبل برای <a href="https://fa-m-wikipedia-org.translate.goog/wiki/%D8%A7%D8%B9%D8%AF%D8%A7%D8%AF_%D9%81%D8%B1%D8%AF?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="اعداد فرد">اعداد فرد</a> به صورت زیر تعریف میشود</p> <p><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 (2k-1)!!=\prod _{i=1}^{k}(2i-1).}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false"> ( </mo> <mn> 2 </mn> <mi> k </mi> <mo> −<!-- − --> </mo> <mn> 1 </mn> <mo stretchy="false"> ) </mo> <mo> ! </mo> <mo> ! </mo> <mo> = </mo> <munderover> <mo> ∏<!-- ∏ --> </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </munderover> <mo stretchy="false"> ( </mo> <mn> 2 </mn> <mi> i </mi> <mo> −<!-- − --> </mo> <mn> 1 </mn> <mo stretchy="false"> ) </mo> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle (2k-1)!!=\prod _{i=1}^{k}(2i-1).} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7833d57f096490033a42e3cf703437c84c6b09fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:23.971ex; height:7.343ex;" alt="{\displaystyle (2k-1)!!=\prod _{i=1}^{k}(2i-1).}"> </noscript><span class="lazy-image-placeholder" style="width: 23.971ex;height: 7.343ex;vertical-align: -3.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7833d57f096490033a42e3cf703437c84c6b09fd" data-alt="{\displaystyle (2k-1)!!=\prod _{i=1}^{k}(2i-1).}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></p> <p>برای مثال</p> <p><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 9!!=1.3.5.7.9}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn> 9 </mn> <mo> ! </mo> <mo> ! </mo> <mo> = </mo> <mn> 1.3.5.7.9 </mn> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle 9!!=1.3.5.7.9} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/efcb32c8ff8651f857f8f9931f66507ea4b36273" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:13.954ex; height:2.176ex;" alt="{\displaystyle 9!!=1.3.5.7.9}"> </noscript><span class="lazy-image-placeholder" style="width: 13.954ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/efcb32c8ff8651f857f8f9931f66507ea4b36273" data-alt="{\displaystyle 9!!=1.3.5.7.9}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></p> <p><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 19!!=1.3.5.7.9.11.13.15.17.19}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn> 19 </mn> <mo> ! </mo> <mo> ! </mo> <mo> = </mo> <mn> 1.3.5.7.9.11.13.15.17.19 </mn> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle 19!!=1.3.5.7.9.11.13.15.17.19} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e2b2f6caf66878c6ef5ef61d4c1945febe36c3d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:29.975ex; height:2.176ex;" alt="{\displaystyle 19!!=1.3.5.7.9.11.13.15.17.19}"> </noscript><span class="lazy-image-placeholder" style="width: 29.975ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e2b2f6caf66878c6ef5ef61d4c1945febe36c3d" data-alt="{\displaystyle 19!!=1.3.5.7.9.11.13.15.17.19}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></p> <p>فاکتوریل دوبل برای <a href="https://fa-m-wikipedia-org.translate.goog/wiki/%D8%A7%D8%B9%D8%AF%D8%A7%D8%AF_%D8%B2%D9%88%D8%AC?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="اعداد زوج">اعداد زوج</a> به صورت زیر تعریف میشود</p> <p><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 (2k)!!=\prod _{i=1}^{k}(2i)=2^{k}k!.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false"> ( </mo> <mn> 2 </mn> <mi> k </mi> <mo stretchy="false"> ) </mo> <mo> ! </mo> <mo> ! </mo> <mo> = </mo> <munderover> <mo> ∏<!-- ∏ --> </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </munderover> <mo stretchy="false"> ( </mo> <mn> 2 </mn> <mi> i </mi> <mo stretchy="false"> ) </mo> <mo> = </mo> <msup> <mn> 2 </mn> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msup> <mi> k </mi> <mo> ! </mo> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle (2k)!!=\prod _{i=1}^{k}(2i)=2^{k}k!.} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2213d08a264dff5e5e613fe75aca27ad4c831224" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:23.173ex; height:7.343ex;" alt="{\displaystyle (2k)!!=\prod _{i=1}^{k}(2i)=2^{k}k!.}"> </noscript><span class="lazy-image-placeholder" style="width: 23.173ex;height: 7.343ex;vertical-align: -3.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2213d08a264dff5e5e613fe75aca27ad4c831224" data-alt="{\displaystyle (2k)!!=\prod _{i=1}^{k}(2i)=2^{k}k!.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></p> <p>برای مثال</p> <p><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 10!!=2.4.6.8.10}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn> 10 </mn> <mo> ! </mo> <mo> ! </mo> <mo> = </mo> <mn> 2.4.6.8.10 </mn> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle 10!!=2.4.6.8.10} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a24f97f87ab4e23d83348a23af848ff63a37b8e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:16.279ex; height:2.176ex;" alt="{\displaystyle 10!!=2.4.6.8.10}"> </noscript><span class="lazy-image-placeholder" style="width: 16.279ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a24f97f87ab4e23d83348a23af848ff63a37b8e" data-alt="{\displaystyle 10!!=2.4.6.8.10}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></p> <p><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 20!!=2.4.6.8.10.12.14.16.18.20}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn> 20 </mn> <mo> ! </mo> <mo> ! </mo> <mo> = </mo> <mn> 2.4.6.8.10.12.14.16.18.20 </mn> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle 20!!=2.4.6.8.10.12.14.16.18.20} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c9d41cf13153e281abe40117e126f3cf773636f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:31.138ex; height:2.176ex;" alt="{\displaystyle 20!!=2.4.6.8.10.12.14.16.18.20}"> </noscript><span class="lazy-image-placeholder" style="width: 31.138ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c9d41cf13153e281abe40117e126f3cf773636f" data-alt="{\displaystyle 20!!=2.4.6.8.10.12.14.16.18.20}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></p> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(2)"> <span class="indicator mf-icon mf-icon-expand mf-icon--small"></span> <h2 id="خواص"><span id=".D8.AE.D9.88.D8.A7.D8.B5"></span>خواص</h2><span class="mw-editsection"> <a role="button" href="https://fa-m-wikipedia-org.translate.goog/w/index.php?title=%D9%81%D8%A7%DA%A9%D8%AA%D9%88%D8%B1%DB%8C%D9%84_%D8%AF%D9%88%D8%A8%D9%84&action=edit&section=2&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="ویرایش بخش: خواص" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>ویرایش</span> </a> </span> </div> <section class="mf-section-2 collapsible-block" id="mf-section-2"> <p><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!!=n\times (n-2)!!}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> n </mi> <mo> ! </mo> <mo> ! </mo> <mo> = </mo> <mi> n </mi> <mo> ×<!-- × --> </mo> <mo stretchy="false"> ( </mo> <mi> n </mi> <mo> −<!-- − --> </mo> <mn> 2 </mn> <mo stretchy="false"> ) </mo> <mo> ! </mo> <mo> ! </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle n!!=n\times (n-2)!!} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5981330b4c21bfe4834d3afff0b140e22b2a017" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.523ex; height:2.843ex;" alt="{\displaystyle n!!=n\times (n-2)!!}"> </noscript><span class="lazy-image-placeholder" style="width: 18.523ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5981330b4c21bfe4834d3afff0b140e22b2a017" data-alt="{\displaystyle n!!=n\times (n-2)!!}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></p> <p><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 (2k)!!=2^{k}k!.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false"> ( </mo> <mn> 2 </mn> <mi> k </mi> <mo stretchy="false"> ) </mo> <mo> ! </mo> <mo> ! </mo> <mo> = </mo> <msup> <mn> 2 </mn> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msup> <mi> k </mi> <mo> ! </mo> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle (2k)!!=2^{k}k!.} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/660edb8dc3c7d5f72722a820d49d7e05559cf856" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.331ex; height:3.176ex;" alt="{\displaystyle (2k)!!=2^{k}k!.}"> </noscript><span class="lazy-image-placeholder" style="width: 13.331ex;height: 3.176ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/660edb8dc3c7d5f72722a820d49d7e05559cf856" data-alt="{\displaystyle (2k)!!=2^{k}k!.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></p> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(3)"> <span class="indicator mf-icon mf-icon-expand mf-icon--small"></span> <h2 id="منابع"><span id=".D9.85.D9.86.D8.A7.D8.A8.D8.B9"></span>منابع</h2><span class="mw-editsection"> <a role="button" href="https://fa-m-wikipedia-org.translate.goog/w/index.php?title=%D9%81%D8%A7%DA%A9%D8%AA%D9%88%D8%B1%DB%8C%D9%84_%D8%AF%D9%88%D8%A8%D9%84&action=edit&section=3&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="ویرایش بخش: منابع" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>ویرایش</span> </a> </span> </div> <section class="mf-section-3 collapsible-block" id="mf-section-3"> <style data-mw-deduplicate="TemplateStyles:r37199623">.mw-parser-output .reflist{font-size:90%;margin-bottom:0.5em;list-style-type:decimal}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style> <div class="reflist"> </div> <p>مشارکتکنندگان ویکیپدیا. «<bdi><span class="plainlinks"><a class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://en.wikipedia.org/wiki/Double_factorial">Double factorial</a></span></bdi>». در <i><a href="https://fa-m-wikipedia-org.translate.goog/wiki/%D9%88%DB%8C%DA%A9%DB%8C%E2%80%8C%D9%BE%D8%AF%DB%8C%D8%A7%DB%8C_%D8%A7%D9%86%DA%AF%D9%84%DB%8C%D8%B3%DB%8C?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="ویکیپدیای انگلیسی">دانشنامهٔ ویکیپدیای انگلیسی</a></i>.</p> <style data-mw-deduplicate="TemplateStyles:r37527932">.mw-parser-output .asbox{position:relative;overflow:hidden}.mw-parser-output .asbox table{background:transparent}.mw-parser-output .asbox p{margin:0}.mw-parser-output .asbox p+p{margin-top:0.25em}.mw-parser-output .asbox-body{font-style:italic}.mw-parser-output .asbox-note{font-size:smaller}.mw-parser-output .asbox .navbar{position:absolute;top:-0.75em;left:1em;display:none}</style> <div role="note" class="metadata plainlinks asbox stub"> <table role="presentation"> <tbody> <tr class="noresize"> <td style="padding-left:0.4em"><span typeof="mw:File"><a href="https://fa-m-wikipedia-org.translate.goog/wiki/%D9%BE%D8%B1%D9%88%D9%86%D8%AF%D9%87:E-to-the-i-pi.svg?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-file-description"> <noscript> <img alt="آیکون خرد" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/35/E-to-the-i-pi.svg/35px-E-to-the-i-pi.svg.png" decoding="async" width="35" height="31" class="mw-file-element" data-file-width="512" data-file-height="453"> </noscript><span class="lazy-image-placeholder" style="width: 35px;height: 31px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/3/35/E-to-the-i-pi.svg/35px-E-to-the-i-pi.svg.png" data-alt="آیکون خرد" data-width="35" data-height="31" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/35/E-to-the-i-pi.svg/53px-E-to-the-i-pi.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/35/E-to-the-i-pi.svg/70px-E-to-the-i-pi.svg.png 2x" data-class="mw-file-element"> </span></a></span></td> <td><p class="asbox-body">این یک <a href="https://fa-m-wikipedia-org.translate.goog/wiki/%D9%88%DB%8C%DA%A9%DB%8C%E2%80%8C%D9%BE%D8%AF%DB%8C%D8%A7:%D8%AE%D8%B1%D8%AF?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="ویکیپدیا:خرد">مقالهٔ خرد</a> <a href="https://fa-m-wikipedia-org.translate.goog/wiki/%D8%B1%DB%8C%D8%A7%D8%B6%DB%8C%D8%A7%D8%AA?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="ریاضیات">ریاضیات</a> است. میتوانید با <a class="external text" 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href="https://fa-m-wikipedia-org.translate.goog/w/index.php?title=%D9%81%D8%A7%DA%A9%D8%AA%D9%88%D8%B1%DB%8C%D9%84_%D8%AF%D9%88%D8%A8%D9%84&action=history&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB"> <div class="post-content last-modified-bar__content"><span class="minerva-icon minerva-icon-size-medium minerva-icon--modified-history"></span> <span class="last-modified-bar__text modified-enhancement" data-user-name="Mo.karimiyan" data-user-gender="unknown" data-timestamp="1543684804"> <span>آخرینبار در ۱ دسامبر ۲۰۱۸، ۱۷:۲۰ ویرایش شده است</span> </span> <span class="minerva-icon minerva-icon-size-small minerva-icon--expand"></span> </div></a> <div class="post-content footer-content"> <div id="mw-data-after-content"> <div class="read-more-container"></div> </div> <div id="p-lang"> <h4>زبانها</h4> <section> <ul id="p-variants" class="minerva-languages"></ul> <ul class="minerva-languages"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ar.wikipedia.org/wiki/%25D8%25B9%25D8%25A7%25D9%2585%25D9%2584%25D9%258A_%25D8%25AB%25D9%2586%25D8%25A7%25D8%25A6%25D9%258A" title="عاملي ثنائي–عربی" lang="ar" hreflang="ar" data-title="عاملي ثنائي" data-language-autonym="العربية" data-language-local-name="عربی" class="interlanguage-link-target"><span>العربية</span></a></li> <li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ca.wikipedia.org/wiki/Doble_factorial" title="Doble factorial–کاتالان" lang="ca" hreflang="ca" data-title="Doble factorial" data-language-autonym="Català" data-language-local-name="کاتالان" class="interlanguage-link-target"><span>Català</span></a></li> <li class="interlanguage-link interwiki-en mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://en.wikipedia.org/wiki/Double_factorial" title="Double factorial–انگلیسی" lang="en" hreflang="en" data-title="Double factorial" data-language-autonym="English" data-language-local-name="انگلیسی" class="interlanguage-link-target"><span>English</span></a></li> <li class="interlanguage-link interwiki-es mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://es.wikipedia.org/wiki/Doble_factorial" title="Doble factorial–اسپانیایی" lang="es" hreflang="es" data-title="Doble factorial" data-language-autonym="Español" data-language-local-name="اسپانیایی" class="interlanguage-link-target"><span>Español</span></a></li> <li class="interlanguage-link interwiki-et mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://et.wikipedia.org/wiki/Kahekordne_faktoriaal" title="Kahekordne faktoriaal–استونیایی" lang="et" hreflang="et" data-title="Kahekordne faktoriaal" data-language-autonym="Eesti" data-language-local-name="استونیایی" class="interlanguage-link-target"><span>Eesti</span></a></li> <li class="interlanguage-link interwiki-he mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://he.wikipedia.org/wiki/%25D7%25A2%25D7%25A6%25D7%25A8%25D7%25AA_%25D7%259B%25D7%25A4%25D7%2595%25D7%259C%25D7%2594" title="עצרת כפולה–عبری" lang="he" hreflang="he" data-title="עצרת כפולה" data-language-autonym="עברית" data-language-local-name="عبری" class="interlanguage-link-target"><span>עברית</span></a></li> <li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ja.wikipedia.org/wiki/%25E4%25BA%258C%25E9%2587%258D%25E9%259A%258E%25E4%25B9%2597" title="二重階乗–ژاپنی" lang="ja" hreflang="ja" data-title="二重階乗" data-language-autonym="日本語" data-language-local-name="ژاپنی" class="interlanguage-link-target"><span>日本語</span></a></li> <li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://nl.wikipedia.org/wiki/Dubbelfaculteit" title="Dubbelfaculteit–هلندی" lang="nl" hreflang="nl" data-title="Dubbelfaculteit" data-language-autonym="Nederlands" data-language-local-name="هلندی" class="interlanguage-link-target"><span>Nederlands</span></a></li> <li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://pt.wikipedia.org/wiki/Duplo_fatorial" title="Duplo fatorial–پرتغالی" lang="pt" hreflang="pt" data-title="Duplo fatorial" data-language-autonym="Português" data-language-local-name="پرتغالی" class="interlanguage-link-target"><span>Português</span></a></li> <li class="interlanguage-link interwiki-ru badge-Q70894304 mw-list-item" title=""><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ru.wikipedia.org/wiki/%25D0%2594%25D0%25B2%25D0%25BE%25D0%25B9%25D0%25BD%25D0%25BE%25D0%25B9_%25D1%2584%25D0%25B0%25D0%25BA%25D1%2582%25D0%25BE%25D1%2580%25D0%25B8%25D0%25B0%25D0%25BB" title="Двойной факториал–روسی" lang="ru" hreflang="ru" data-title="Двойной факториал" data-language-autonym="Русский" data-language-local-name="روسی" class="interlanguage-link-target"><span>Русский</span></a></li> <li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://simple.wikipedia.org/wiki/Double_factorial" title="Double factorial–انگلیسی ساده" lang="en-simple" hreflang="en-simple" data-title="Double factorial" data-language-autonym="Simple English" data-language-local-name="انگلیسی ساده" class="interlanguage-link-target"><span>Simple English</span></a></li> <li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://sk.wikipedia.org/wiki/Dvojit%25C3%25BD_faktori%25C3%25A1l" title="Dvojitý faktoriál–اسلواکی" lang="sk" hreflang="sk" data-title="Dvojitý faktoriál" data-language-autonym="Slovenčina" data-language-local-name="اسلواکی" class="interlanguage-link-target"><span>Slovenčina</span></a></li> <li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://sv.wikipedia.org/wiki/Semifakultet" title="Semifakultet–سوئدی" lang="sv" hreflang="sv" data-title="Semifakultet" data-language-autonym="Svenska" data-language-local-name="سوئدی" class="interlanguage-link-target"><span>Svenska</span></a></li> <li class="interlanguage-link interwiki-uk badge-Q70893996 mw-list-item" title=""><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://uk.wikipedia.org/wiki/%25D0%259F%25D0%25BE%25D0%25B4%25D0%25B2%25D1%2596%25D0%25B9%25D0%25BD%25D0%25B8%25D0%25B9_%25D1%2584%25D0%25B0%25D0%25BA%25D1%2582%25D0%25BE%25D1%2580%25D1%2596%25D0%25B0%25D0%25BB" title="Подвійний факторіал–اوکراینی" lang="uk" hreflang="uk" data-title="Подвійний факторіал" data-language-autonym="Українська" data-language-local-name="اوکراینی" class="interlanguage-link-target"><span>Українська</span></a></li> <li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://zh-yue.wikipedia.org/wiki/%25E9%259B%2599%25E9%259A%258E%25E4%25B9%2598" title="雙階乘–کانتونی" lang="yue" hreflang="yue" data-title="雙階乘" data-language-autonym="粵語" data-language-local-name="کانتونی" class="interlanguage-link-target"><span>粵語</span></a></li> </ul> </section> </div> <div class="minerva-footer-logo"> <img 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