Max[Table[Binomial[n, k], {k, 0, n}]] == Binomial[n - l, l] /; l == Floor[(1/10) (5 n + 7 - Sqrt[5 n^2 + 10 n + 9])]

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Max", "[", RowBox[List["Table", "[", RowBox[List[RowBox[List["Binomial", "[", RowBox[List["n", ",", "k"]], "]"]], ",", RowBox[List["{", RowBox[List["k", ",", "0", ",", "n"]], "}"]]]], "]"]], "]"]], "\[Equal]", RowBox[List["Binomial", "[", RowBox[List[RowBox[List["n", "-", "l"]], ",", "l"]], "]"]]]], "/;", RowBox[List["l", "\[Equal]", RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "10"], " ", RowBox[List["(", RowBox[List[RowBox[List["5", " ", "n"]], "+", "7", "-", SqrtBox[RowBox[List[RowBox[List["5", " ", SuperscriptBox["n", "2"]]], "+", RowBox[List["10", " ", "n"]], "+", "9"]]]]], ")"]]]], "]"]]]]]]]]

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <mi> max </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <semantics> <mrow> <mo> ( </mo> <mtable> <mtr> <mtd> <mi> n </mi> </mtd> </mtr> <mtr> <mtd> <mn> 0 </mn> </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;0&quot;, Identity, Rule[Editable, True]]]]], &quot;)&quot;]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] </annotation> </semantics> <mo> , </mo> <semantics> <mrow> <mo> ( </mo> <mtable> <mtr> <mtd> <mi> n </mi> </mtd> </mtr> <mtr> <mtd> <mn> 1 </mn> </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;1&quot;, Identity, Rule[Editable, True]]]]], &quot;)&quot;]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] </annotation> </semantics> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <semantics> <mrow> <mo> ( </mo> <mtable> <mtr> <mtd> <mi> n </mi> </mtd> </mtr> <mtr> <mtd> <mi> n </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;n&quot;, Identity, Rule[Editable, True]]]]], &quot;)&quot;]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] </annotation> </semantics> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <semantics> <mrow> <mo> ( </mo> <mtable> <mtr> <mtd> <mrow> <mi> n </mi> <mo> - </mo> <mi> l </mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mi> l </mi> </mtd> </mtr> </mtable> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;(&quot;, GridBox[List[List[TagBox[RowBox[List[&quot;n&quot;, &quot;-&quot;, &quot;l&quot;]], Identity, Rule[Editable, True]]], List[TagBox[&quot;l&quot;, Identity, Rule[Editable, True]]]]], &quot;)&quot;]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] </annotation> </semantics> </mrow> <mo> /; </mo> <mrow> <mi> l </mi> <mo> &#10869; </mo> <mrow> <mo> &#8970; </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 10 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 5 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mo> - </mo> <msqrt> <mrow> <mrow> <mn> 5 </mn> <mo> &#8290; </mo> <msup> <mi> n </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 10 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mo> + </mo> <mn> 9 </mn> </mrow> </msqrt> <mo> + </mo> <mn> 7 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> &#8971; </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <max /> <apply> <ci> Binomial </ci> <ci> n </ci> <cn type='integer'> 0 </cn> </apply> <apply> <ci> Binomial </ci> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> <ci> &#8230; </ci> <apply> <ci> Binomial </ci> <ci> n </ci> <ci> n </ci> </apply> </apply> <apply> <ci> Binomial </ci> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> l </ci> </apply> </apply> <ci> l </ci> </apply> </apply> <apply> <eq /> <ci> l </ci> <apply> <floor /> <apply> <times /> <cn type='rational'> 1 <sep /> 10 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 5 </cn> <ci> n </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 5 </cn> <apply> <power /> <ci> n </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 10 </cn> <ci> n </ci> </apply> <cn type='integer'> 9 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> 7 </cn> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Max", "[", RowBox[List["Table", "[", RowBox[List[RowBox[List["Binomial", "[", RowBox[List["n_", ",", "k_"]], "]"]], ",", RowBox[List["{", RowBox[List["k_", ",", "0", ",", "n_"]], "}"]]]], "]"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["Binomial", "[", RowBox[List[RowBox[List["n", "-", "l"]], ",", "l"]], "]"]], "/;", RowBox[List["l", "\[Equal]", RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "10"], " ", RowBox[List["(", RowBox[List[RowBox[List["5", " ", "n"]], "+", "7", "-", SqrtBox[RowBox[List[RowBox[List["5", " ", SuperscriptBox["n", "2"]]], "+", RowBox[List["10", " ", "n"]], "+", "9"]]]]], ")"]]]], "]"]]]]]]]]]]

2001-10-29