User:Gerry Martens

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<link rel="stylesheet" href="/w/load.php?debug=false&lang=en&modules=ext.math.styles%7Cmediawiki.legacy.commonPrint%2Cshared%7Cmediawiki.sectionAnchor%7Cmediawiki.skinning.content.externallinks%7Cmediawiki.skinning.interface%7Cskins.monobook.styles&only=styles&skin=monobook"/> <script async="" src="/w/load.php?debug=false&lang=en&modules=startup&only=scripts&skin=monobook"></script>  <meta name="ResourceLoaderDynamicStyles" content=""/> <meta name="generator" content="MediaWiki 1.30.0"/> <link rel="shortcut icon" href="/favicon.ico"/> <link rel="search" type="application/opensearchdescription+xml" href="/w/opensearch_desc.php" title="OeisWiki (en)"/> <link rel="EditURI" type="application/rsd+xml" href="https://oeis.org/w/api.php?action=rsd"/> <link rel="license" href="/wiki/The_OEIS_End-User_License_Agreement"/> <link rel="alternate" type="application/atom+xml" title="OeisWiki Atom feed" href="/w/index.php?title=Special:RecentChanges&feed=atom"/>  </head> <body class="mediawiki ltr sitedir-ltr mw-hide-empty-elt ns-2 ns-subject page-User_Gerry_Martens rootpage-User_Gerry_Martens skin-monobook action-view"><div id="globalWrapper"> <div id="column-content"> <div id="content" class="mw-body" role="main"> <a id="top"></a> <div id="siteNotice" class="mw-body-content"><div id="localNotice" lang="en" dir="ltr"><div class="mw-parser-output"><p>This site is supported by donations to <a rel="nofollow" class="external text" href="http://oeisf.org/">The OEIS Foundation</a>. </p></div></div></div> <div class="mw-indicators mw-body-content"> </div> <h1 id="firstHeading" class="firstHeading" lang="en">User:Gerry Martens</h1> <div id="bodyContent" class="mw-body-content"> <div id="siteSub">From OeisWiki</div> <div id="contentSub"></div> <div id="jump-to-nav" class="mw-jump">Jump to: <a href="#column-one">navigation</a>, <a href="#searchInput">search</a></div>  <div id="mw-content-text" lang="en" dir="ltr" class="mw-content-ltr"><div class="mw-parser-output"><div style="width:800px;text-align:justify;"> <div id="toc" class="toc"><div class="toctitle"><h2>Contents</h2></div> <ul> <li class="toclevel-1"><a href="#Binomial_identities_:"><span class="tocnumber">1</span> <span class="toctext">Binomial identities :</span></a> <ul> <li class="toclevel-2"><a href="#A000984_The_Central_Binomial_Coefficient_:"><span class="tocnumber">1.1</span> <span class="toctext">A000984 The Central Binomial Coefficient :</span></a></li> <li class="toclevel-2"><a href="#The_Binomial_Coefficient_and_its_square_:"><span class="tocnumber">1.2</span> <span class="toctext">The Binomial Coefficient and its square :</span></a></li> <li class="toclevel-2"><a href="#The_Binomial_Coefficient_with_offset_and_its_square_:"><span class="tocnumber">1.3</span> <span class="toctext">The Binomial Coefficient with offset and its square :</span></a></li> </ul> </li> </ul> </div> <h1 style="color: blue;"><span class="mw-headline" id="Binomial_identities_:">Binomial identities :</span></h1> <ul> <li> In case you spot any related sequences or have comments, you can post them on my <a href="/w/index.php?title=User_talk:Gerry_Martens&action=edit&redlink=1" class="new" title="User talk:Gerry Martens (page does not exist)">user_talk_page</a>.</li> </ul> <h2 style="color: blue;"><span class="mw-headline" id="A000984_The_Central_Binomial_Coefficient_:"><a href="http://oeis.org/A000984">A000984</a> The Central Binomial Coefficient :</span></h2> <p>Reviewing some OEIS binomial related sequences one notices the following form for certain p and q : </p> <div style="margin-left: 20px; width:560px; margin-bottom:16px"><table><tr> <td style="background-color: #ADD8E6;color: #820000;font-weight: 600;font-size: 1.0em; border: inset 1pt red;padding: 0.8em;"> <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 {\begin{aligned}a(n)&=\left(-q^{2}\right)^{n}{\binom {\frac {p}{q}}{n}}=q^{2n}{\binom {n-1-{\frac {p}{q}}}{-1-{\frac {p}{q}}}}={\frac {q^{2n}}{n!}}\left(-{\frac {p}{q}}\right)_{n}={\frac {q^{n}}{n!}}\prod _{k=0}^{n-1}{qk-p}=q^{2n}C_{n}^{\{-{\frac {p}{2q}}\}}{(1)}=\left[x^{n}\right]\left(1-q^{2}x\right)^{\frac {p}{q}}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>a</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</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"> <mfrac> <mi>p</mi> <mi>q</mi> </mfrac> <mi>n</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo>=</mo> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</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> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>p</mi> <mi>q</mi> </mfrac> </mrow> </mrow> <mrow> <mo>−</mo> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>p</mi> <mi>q</mi> </mfrac> </mrow> </mrow> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msup> <mrow> <mi>n</mi> <mo>!</mo> </mrow> </mfrac> </mrow> <msub> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>p</mi> <mi>q</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mrow> <mi>n</mi> <mo>!</mo> </mrow> </mfrac> </mrow> <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"> <mi>q</mi> <mi>k</mi> <mo>−</mo> <mi>p</mi> </mrow> <mo>=</mo> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msup> <msubsup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo fence="false" stretchy="false">{</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>p</mi> <mrow> <mn>2</mn> <mi>q</mi> </mrow> </mfrac> </mrow> <mo fence="false" stretchy="false">}</mo> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mrow> <mo>[</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>]</mo> </mrow> <msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>p</mi> <mi>q</mi> </mfrac> </mrow> </msup> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}a(n)&=\left(-q^{2}\right)^{n}{\binom {\frac {p}{q}}{n}}=q^{2n}{\binom {n-1-{\frac {p}{q}}}{-1-{\frac {p}{q}}}}={\frac {q^{2n}}{n!}}\left(-{\frac {p}{q}}\right)_{n}={\frac {q^{n}}{n!}}\prod _{k=0}^{n-1}{qk-p}=q^{2n}C_{n}^{\{-{\frac {p}{2q}}\}}{(1)}=\left[x^{n}\right]\left(1-q^{2}x\right)^{\frac {p}{q}}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://en.wikipedia.org/api/rest_v1/media/math/render/svg/124d5081dab841229f4f6a9c89cb45931c85c91a" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -3.338ex; width:107.014ex; height:7.843ex;" alt="{\displaystyle {\begin{aligned}a(n)&=\left(-q^{2}\right)^{n}{\binom {\frac {p}{q}}{n}}=q^{2n}{\binom {n-1-{\frac {p}{q}}}{-1-{\frac {p}{q}}}}={\frac {q^{2n}}{n!}}\left(-{\frac {p}{q}}\right)_{n}={\frac {q^{n}}{n!}}\prod _{k=0}^{n-1}{qk-p}=q^{2n}C_{n}^{\{-{\frac {p}{2q}}\}}{(1)}=\left[x^{n}\right]\left(1-q^{2}x\right)^{\frac {p}{q}}\end{aligned}}}" /></span></td> </tr></table></div> <p>By assigning p=1 and q=-2 the sequence a(n) is the central binomial coefficient and one obtains the following identity: </p> <div style="margin-left: 20px; width:560px; margin-bottom:16px"><table><tr> <td style="background-color: #ADD8E6;color: #820000;font-weight: 600;font-size: 1.0em; border: inset 1pt red;padding: 0.8em;"> <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 {\begin{aligned}{\binom {2n}{n}}=\left(-4\right)^{n}{\binom {-{\frac {1}{2}}}{n}}=4^{n}{\binom {n-{\frac {1}{2}}}{-{\frac {1}{2}}}}={\frac {4^{n}}{n!}}\left({\frac {1}{2}}\right)_{n}={\frac {(-2)^{n}}{n!}}\prod _{k=0}^{n-1}{-2k-1}=4^{n}C_{n}^{\{{\frac {1}{4}}\}}{(1)}=\left[x^{n}\right]{\frac {1}{\sqrt {1-4x}}}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mn>2</mn> <mi>n</mi> </mrow> <mi>n</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mn>4</mn> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</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> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mrow> <mi>n</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo>=</mo> <msup> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</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> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mrow> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mrow> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mrow> <mi>n</mi> <mo>!</mo> </mrow> </mfrac> </mrow> <msub> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mn>2</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> <mrow> <mi>n</mi> <mo>!</mo> </mrow> </mfrac> </mrow> <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"> <mo>−</mo> <mn>2</mn> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> <mo>=</mo> <msup> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <msubsup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo fence="false" stretchy="false">{</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <mo fence="false" stretchy="false">}</mo> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mrow> <mo>[</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>1</mn> <mo>−</mo> <mn>4</mn> <mi>x</mi> </msqrt> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\binom {2n}{n}}=\left(-4\right)^{n}{\binom {-{\frac {1}{2}}}{n}}=4^{n}{\binom {n-{\frac {1}{2}}}{-{\frac {1}{2}}}}={\frac {4^{n}}{n!}}\left({\frac {1}{2}}\right)_{n}={\frac {(-2)^{n}}{n!}}\prod _{k=0}^{n-1}{-2k-1}=4^{n}C_{n}^{\{{\frac {1}{4}}\}}{(1)}=\left[x^{n}\right]{\frac {1}{\sqrt {1-4x}}}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://en.wikipedia.org/api/rest_v1/media/math/render/svg/0aeb888aa0efec07aa4be2ca8e5d2951c47be26c" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -3.338ex; width:102.663ex; height:7.676ex;" alt="{\displaystyle {\begin{aligned}{\binom {2n}{n}}=\left(-4\right)^{n}{\binom {-{\frac {1}{2}}}{n}}=4^{n}{\binom {n-{\frac {1}{2}}}{-{\frac {1}{2}}}}={\frac {4^{n}}{n!}}\left({\frac {1}{2}}\right)_{n}={\frac {(-2)^{n}}{n!}}\prod _{k=0}^{n-1}{-2k-1}=4^{n}C_{n}^{\{{\frac {1}{4}}\}}{(1)}=\left[x^{n}\right]{\frac {1}{\sqrt {1-4x}}}\end{aligned}}}" /></span></td> </tr></table></div> <h2 style="color: blue;"><span class="mw-headline" id="The_Binomial_Coefficient_and_its_square_:">The Binomial Coefficient and its square :</span></h2> <div style="margin-left: 20px; width:560px; margin-bottom:16px"><table><tr> <td style="background-color: #ADD8E6;color: #820000;font-weight: 600;font-size: 1.0em; border: inset 1pt red;padding: 0.8em;"> <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 {\binom {k}{i}}={\binom {k}{i}}^{2}+2\sum _{j=1}^{i}{(-1)^{j}{\binom {k}{i-j}}{\binom {k}{i+j}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>k</mi> <mi>i</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>k</mi> <mi>i</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</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"> <mi>k</mi> <mrow> <mi>i</mi> <mo>−</mo> <mi>j</mi> </mrow> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>k</mi> <mrow> <mi>i</mi> <mo>+</mo> <mi>j</mi> </mrow> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\binom {k}{i}}={\binom {k}{i}}^{2}+2\sum _{j=1}^{i}{(-1)^{j}{\binom {k}{i-j}}{\binom {k}{i+j}}}}</annotation> </semantics> </math></span><img src="https://en.wikipedia.org/api/rest_v1/media/math/render/svg/f06b0aecf7d2dba3da16641d41e2999976ffb061" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -3.338ex; width:43.283ex; height:7.509ex;" alt="{\displaystyle {\binom {k}{i}}={\binom {k}{i}}^{2}+2\sum _{j=1}^{i}{(-1)^{j}{\binom {k}{i-j}}{\binom {k}{i+j}}}}" /></span></td> </tr></table></div> <ul> <li> It is a little challenging writing the identity this way but the (-1)^j takes care of the sign. <br /> Due to its origin it is more meaningful using the variables (i,j,k). <br /> For the OEIS sequences it is common to replace k by n. </li> </ul> <table cellpadding="10" border="1"><tr> <td style="line-height: 1.5em; background:#ADD8E6; color:black; text-align:center; font-weight:bold">Related <br /> Sequence</td> <td style="line-height: 1.5em; background:#ADD8E6; color:black; text-align:center; font-weight:bold">Name</td> </tr><tr> <td><a href="http://oeis.org/A108958">A108958</a></td> <td>Number of unordered pairs of distinct length-n binary words having the same number of 1's.</td> </tr><tr> <td><a href="http://oeis.org/A054563">A054563</a></td> <td>a(n) = n*(n^2 - 1)*(n + 2)*(n^2 + 4*n + 6)/72.</td> </table> <p><br /> </p> <h2 style="color: blue;"><span class="mw-headline" id="The_Binomial_Coefficient_with_offset_and_its_square_:">The Binomial Coefficient with offset and its square :</span></h2> <div style="margin-left: 20px; width:560px; margin-bottom:16px"><table><tr> <td style="background-color: #ADD8E6;color: #820000;font-weight: 600;font-size: 1.0em; border: inset 1pt red;padding: 0.8em;"> <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 {\begin{aligned}{\binom {n+k}{k}}&={\binom {n+k}{k}}^{2}+2\sum _{i=1}^{k}(-1)^{i}{{\binom {n+k}{k+i}}{\binom {n+k}{k-i}}}\;\;&\forall \;\;{\;k\in \mathbb {N} }\\&={\binom {n+k}{k}}^{2}-2{\binom {n+k}{k+1}}{\binom {n+k}{k-1}}\,_{3}F_{2}(1,1-k,1-n;2+k,2+n;-1)\;\;&\forall \;\;{\;k\in \mathbb {Q} }\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <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> <mi>k</mi> </mrow> <mi>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <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> <mi>k</mi> </mrow> <mi>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <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> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <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> <mi>k</mi> </mrow> <mrow> <mi>k</mi> <mo>+</mo> <mi>i</mi> </mrow> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <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> <mi>k</mi> </mrow> <mrow> <mi>k</mi> <mo>−</mo> <mi>i</mi> </mrow> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> </mrow> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> </mtd> <mtd> <mi mathvariant="normal">∀</mi> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mspace width="thickmathspace" /> <mi>k</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <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> <mi>k</mi> </mrow> <mi>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <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> <mi>k</mi> </mrow> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <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> <mi>k</mi> </mrow> <mrow> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <msub> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>−</mo> <mi>k</mi> <mo>,</mo> <mn>1</mn> <mo>−</mo> <mi>n</mi> <mo>;</mo> <mn>2</mn> <mo>+</mo> <mi>k</mi> <mo>,</mo> <mn>2</mn> <mo>+</mo> <mi>n</mi> <mo>;</mo> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> </mtd> <mtd> <mi mathvariant="normal">∀</mi> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mspace width="thickmathspace" /> <mi>k</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\binom {n+k}{k}}&={\binom {n+k}{k}}^{2}+2\sum _{i=1}^{k}(-1)^{i}{{\binom {n+k}{k+i}}{\binom {n+k}{k-i}}}\;\;&\forall \;\;{\;k\in \mathbb {N} }\\&={\binom {n+k}{k}}^{2}-2{\binom {n+k}{k+1}}{\binom {n+k}{k-1}}\,_{3}F_{2}(1,1-k,1-n;2+k,2+n;-1)\;\;&\forall \;\;{\;k\in \mathbb {Q} }\end{aligned}}}</annotation> </semantics> </math></span><img src="https://en.wikipedia.org/api/rest_v1/media/math/render/svg/b6714274d370863d85a84dff6c0c746bb77131a9" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -6.505ex; width:95.726ex; height:14.176ex;" alt="{\displaystyle {\begin{aligned}{\binom {n+k}{k}}&={\binom {n+k}{k}}^{2}+2\sum _{i=1}^{k}(-1)^{i}{{\binom {n+k}{k+i}}{\binom {n+k}{k-i}}}\;\;&\forall \;\;{\;k\in \mathbb {N} }\\&={\binom {n+k}{k}}^{2}-2{\binom {n+k}{k+1}}{\binom {n+k}{k-1}}\,_{3}F_{2}(1,1-k,1-n;2+k,2+n;-1)\;\;&\forall \;\;{\;k\in \mathbb {Q} }\end{aligned}}}" /></span></td> </tr></table></div> <p><br /> </p> <table cellpadding="10" border="1"><tr> <td style="line-height: 1.5em; background:#ADD8E6; color:black; text-align:center; font-weight:bold">n</td> <td style="line-height: 1.5em; background:#ADD8E6; color:black; text-align:center; font-weight:bold">k</td> <td style="line-height: 1.5em; background:#ADD8E6; color:black; text-align:center; font-weight:bold">Related <br /> Sequence</td> <td style="line-height: 1.5em; background:#ADD8E6; color:black; text-align:center; font-weight:bold">Name</td> </tr><tr> <td>n</td> <td>1</td> <td><a href="http://oeis.org/A000027">A000027</a></td> <td>The positive integers.</td> </tr><tr> <td>n</td> <td>2</td> <td><a href="http://oeis.org/A000217">A000217</a></td> <td>Triangular numbers: a(n) = binomial(n+1,2).</td> </tr><tr> <td>n</td> <td>3</td> <td><a href="http://oeis.org/A000292">A000292</a></td> <td>Tetrahedral (or triangular pyramidal) numbers: a(n) = C(n+2,3).</td> </tr><tr> <td>n</td> <td>4</td> <td><a href="http://oeis.org/A000332">A000332</a></td> <td>Binomial coefficient binomial(n,4)</td> </tr><tr> <td>n</td> <td>5</td> <td><a href="http://oeis.org/A000389">A000389</a></td> <td>Binomial coefficients C(n,5).</td> </tr><tr> <td>n</td> <td>6</td> <td><a href="http://oeis.org/A000579">A000579</a></td> <td>Figurate numbers or binomial coefficients C(n,6).</td> </tr><tr> <td>n</td> <td>7</td> <td><a href="http://oeis.org/A000580">A000580</a></td> <td>a(n) = binomial coefficient C(n,7).</td> </tr><tr> <td>n</td> <td>8</td> <td><a href="http://oeis.org/A000581">A000581</a></td> <td>a(n) = binomial coefficient C(n,8).</td> </tr><tr> <td>n</td> <td>9</td> <td><a href="http://oeis.org/A000582">A000582</a></td> <td>a(n) = binomial coefficient C(n,9).</td> </tr><tr> <td>n</td> <td>10</td> <td><a href="http://oeis.org/A001287">A001287</a></td> <td>a(n) = binomial coefficient C(n,10).</td> </tr><tr> <td>n</td> <td>11</td> <td><a href="http://oeis.org/A001288">A001288</a></td> <td>a(n) = binomial(n,11).</td> </tr><tr> <td>n</td> <td>12</td> <td><a href="http://oeis.org/A010965">A010965</a></td> <td>a(n) = binomial(n,12).</td> </tr><tr> <td>n</td> <td>13</td> <td><a href="http://oeis.org/A010966">A010966</a></td> <td>a(n) = binomial(n,13).</td> </table> <p><br /> <br /> </p> <table cellpadding="10" border="1"><tr> <td style="line-height: 1.5em; background:#ADD8E6; color:black; text-align:center; font-weight:bold">n</td> <td style="line-height: 1.5em; background:#ADD8E6; color:black; text-align:center; font-weight:bold">k</td> <td style="line-height: 1.5em; background:#ADD8E6; color:black; text-align:center; font-weight:bold">Related <br /> Sequence</td> <td style="line-height: 1.5em; background:#ADD8E6; color:black; text-align:center; font-weight:bold">Name</td> </tr><tr> </tr><tr> <td>2n</td> <td>1</td> <td><a href="http://oeis.org/A005408">A005408</a></td> <td>The odd numbers: a(n) = 2*n + 1.</td> </tr><tr> <td>2n</td> <td>2</td> <td><a href="http://oeis.org/A000384">A000384</a></td> <td>Hexagonal numbers : a (n) = n*(2*n - 1) = C(2*n,2).</td> </tr><tr> <td>2n</td> <td>4</td> <td><a href="http://oeis.org/A053134">A053134</a></td> <td>Binomial coefficients C (2*n + 4, 4).</td> </tr><tr> <td>2n</td> <td>6</td> <td><a href="http://oeis.org/A053135">A053135</a></td> <td>Binomial coefficients C (2*n + 6, 6).</td> </tr><tr> <td>2n</td> <td>8</td> <td><a href="http://oeis.org/A053137">A053137</a></td> <td>Binomial coefficients C (2*n + 8, 8).</td> </tr><tr> <td>2n</td> <td>10</td> <td><a href="http://oeis.org/A196789">A196789</a></td> <td>Binomial coefficients C(2*n+10,10).</td> </table>   </div>  </div><div class="printfooter"> Retrieved from "<a dir="ltr" href="https://oeis.org/w/index.php?title=User:Gerry_Martens&oldid=1651732">https://oeis.org/w/index.php?title=User:Gerry_Martens&oldid=1651732</a>"</div> <div id="catlinks" class="catlinks catlinks-allhidden" data-mw="interface"></div>  <div class="visualClear"></div> </div> </div> </div> <div id="column-one"> <h2>Navigation menu</h2> <div id="p-cactions" class="portlet" role="navigation"> <h3>Views</h3> <div class="pBody"> <ul> <li id="ca-nstab-user" class="selected"><a href="/wiki/User:Gerry_Martens" title="View the user page [c]" accesskey="c">User page</a></li> <li id="ca-talk" class="new"><a href="/w/index.php?title=User_talk:Gerry_Martens&action=edit&redlink=1" 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