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Binomial




Mathematica Notation

Traditional Notation









Gamma, Beta, Erf > Binomial[n,k] > Summation > Finite summation





http://functions.wolfram.com/06.03.23.0016.01









  


  








Input Form





Sum[(Binomial[n, r + k] Binomial[n + r + k, k])/(-2)^k, {k, 0, n - r}] == ((-1)^m Binomial[n, m])/2^(2 m) /; (Element[n, Integers] && n >= 0 && Element[r, Integers] && r >= 0 && Element[(n - r)/2, Integers] && m = (n - r)/2)








Standard Form





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MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mtext> </mtext> <mrow> <mrow> <mrow> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow> <mi> n </mi> <mo> - </mo> <mi> r </mi> </mrow> </munderover> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mo> - </mo> <mi> k </mi> </mrow> </msup> <mo> &#8290; </mo> <semantics> <mrow> <mo> ( </mo> <mtable> <mtr> <mtd> <mi> n </mi> </mtd> </mtr> <mtr> <mtd> <mrow> <mi> k </mi> <mo> + </mo> <mi> r </mi> </mrow> </mtd> </mtr> </mtable> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;(&quot;, GridBox[List[List[TagBox[&quot;n&quot;, Identity, Rule[Editable, True]]], List[TagBox[RowBox[List[&quot;k&quot;, &quot;+&quot;, &quot;r&quot;]], Identity, Rule[Editable, True]]]]], &quot;)&quot;]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] </annotation> </semantics> <mo> &#8290; </mo> <semantics> <mrow> <mo> ( </mo> <mtable> <mtr> <mtd> <mrow> <mi> k </mi> <mo> + </mo> <mi> n </mi> <mo> + </mo> <mi> r </mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mi> k </mi> </mtd> </mtr> </mtable> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;(&quot;, GridBox[List[List[TagBox[RowBox[List[&quot;k&quot;, &quot;+&quot;, &quot;n&quot;, &quot;+&quot;, &quot;r&quot;]], Identity, Rule[Editable, True]]], List[TagBox[&quot;k&quot;, Identity, Rule[Editable, True]]]]], &quot;)&quot;]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] </annotation> </semantics> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> m </mi> </msup> <mo> &#8290; </mo> <msup> <mn> 2 </mn> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> &#8290; </mo> <mi> m </mi> </mrow> </msup> <mo> &#8290; </mo> <semantics> <mrow> <mo> ( </mo> <mtable> <mtr> <mtd> <mi> n </mi> </mtd> </mtr> <mtr> <mtd> <mi> m </mi> </mtd> </mtr> </mtable> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;(&quot;, GridBox[List[List[TagBox[&quot;n&quot;, Identity, Rule[Editable, True]]], List[TagBox[&quot;m&quot;, Identity, Rule[Editable, True]]]]], &quot;)&quot;]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] </annotation> </semantics> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> n </mi> <mo> &#8712; </mo> <semantics> <mi> &#8469; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalN]&quot;, Function[Integers]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mi> r </mi> <mo> &#8712; </mo> <semantics> <mi> &#8469; </mi> <annotation encoding='Mathematica'> TagBox[TagBox[&quot;\[DoubleStruckCapitalN]&quot;, Function[Integers]], Function[Integers]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mfrac> <mrow> <mi> n </mi> <mo> - </mo> <mi> r </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> &#8712; </mo> <mi> &#8484; </mi> </mrow> <mo> &#8743; </mo> <mi> m </mi> </mrow> </mrow> <mo> = </mo> <mfrac> <mrow> <mi> n </mi> <mo> - </mo> <mi> r </mi> </mrow> <mn> 2 </mn> </mfrac> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Set </ci> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> r </ci> </apply> </apply> </uplimit> <apply> <times /> <apply> <power /> <cn type='integer'> -2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> </apply> <apply> <ci> Binomial </ci> <ci> n </ci> <apply> <plus /> <ci> k </ci> <ci> r </ci> </apply> </apply> <apply> <ci> Binomial </ci> <apply> <plus /> <ci> k </ci> <ci> n </ci> <ci> r </ci> </apply> <ci> k </ci> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <times /> <cn type='integer'> -2 </cn> <ci> m </ci> </apply> </apply> <apply> <ci> Binomial </ci> <ci> n </ci> <ci> m </ci> </apply> </apply> </apply> <apply> <and /> <apply> <in /> <ci> n </ci> <integers /> </apply> <apply> <in /> <ci> r </ci> <integers /> </apply> <apply> <in /> <apply> <times /> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> r </ci> </apply> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <ci> &#8484; </ci> </apply> <ci> m </ci> </apply> </apply> <apply> <times /> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> r </ci> </apply> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





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Date Added to functions.wolfram.com (modification date)





2002-12-18