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 | | http://functions.wolfram.com/05.14.23.0005.01 | 
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 | | Sum[Binomial[n, k] BernoulliB[k, z] BernoulliB[n - k, w], {k, 0, n}] == 
 n (z + w - 1) BernoulliB[n - 1, z + w] - (n - 1) BernoulliB[n, z + w] | 
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 | | Cell[BoxData[RowBox[List[" ", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], RowBox[List[RowBox[List["Binomial", "[", RowBox[List["n", ",", "k"]], "]"]], RowBox[List["BernoulliB", "[", RowBox[List["k", ",", "z"]], "]"]], RowBox[List["BernoulliB", "[", RowBox[List[RowBox[List["n", "-", "k"]], ",", "w"]], "]"]]]]]], "\[Equal]", RowBox[List[RowBox[List["n", RowBox[List["(", RowBox[List["z", "+", "w", "-", "1"]], ")"]], RowBox[List["BernoulliB", "[", RowBox[List[RowBox[List["n", "-", "1"]], ",", RowBox[List["z", "+", "w"]]]], "]"]]]], "-", RowBox[List[RowBox[List["(", RowBox[List["n", "-", "1"]], ")"]], RowBox[List["BernoulliB", "[", RowBox[List["n", ",", RowBox[List["z", "+", "w"]]]], "]"]]]]]]]]]]]] | 
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   <math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'>  <semantics>  <mrow>  <mrow>  <munderover>  <mo> ∑ </mo>  <mrow>  <mi> k </mi>  <mo> = </mo>  <mn> 0 </mn>  </mrow>  <mi> n </mi>  </munderover>  <mrow>  <semantics>  <mrow>  <mo> ( </mo>  <mtable>  <mtr>  <mtd>  <mi> n </mi>  </mtd>  </mtr>  <mtr>  <mtd>  <mi> k </mi>  </mtd>  </mtr>  </mtable>  <mo> ) </mo>  </mrow>  <annotation encoding='Mathematica'> TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity, Rule[Editable, True]]], List[TagBox["k", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] </annotation>  </semantics>  <mo> ⁢ </mo>  <mrow>  <msub>  <semantics>  <mi> B </mi>  <annotation encoding='Mathematica'> TagBox["B", BernoulliB] </annotation>  </semantics>  <mi> k </mi>  </msub>  <mo> ( </mo>  <mi> z </mi>  <mo> ) </mo>  </mrow>  <mo> ⁢ </mo>  <mrow>  <msub>  <semantics>  <mi> B </mi>  <annotation encoding='Mathematica'> TagBox["B", BernoulliB] </annotation>  </semantics>  <mrow>  <mi> n </mi>  <mo> - </mo>  <mi> k </mi>  </mrow>  </msub>  <mo> ( </mo>  <mi> w </mi>  <mo> ) </mo>  </mrow>  </mrow>  </mrow>  <mo> ⩵ </mo>  <mrow>  <mrow>  <mi> n </mi>  <mo> ⁢ </mo>  <mrow>  <mo> ( </mo>  <mrow>  <mi> w </mi>  <mo> + </mo>  <mi> z </mi>  <mo> - </mo>  <mn> 1 </mn>  </mrow>  <mo> ) </mo>  </mrow>  <mo> ⁢ </mo>  <mrow>  <msub>  <semantics>  <mi> B </mi>  <annotation encoding='Mathematica'> TagBox["B", BernoulliB] </annotation>  </semantics>  <mrow>  <mi> n </mi>  <mo> - </mo>  <mn> 1 </mn>  </mrow>  </msub>  <mo> ( </mo>  <mrow>  <mi> w </mi>  <mo> + </mo>  <mi> z </mi>  </mrow>  <mo> ) </mo>  </mrow>  </mrow>  <mo> - </mo>  <mrow>  <mrow>  <mo> ( </mo>  <mrow>  <mi> n </mi>  <mo> - </mo>  <mn> 1 </mn>  </mrow>  <mo> ) </mo>  </mrow>  <mo> ⁢ </mo>  <mrow>  <msub>  <semantics>  <mi> B </mi>  <annotation encoding='Mathematica'> TagBox["B", BernoulliB] </annotation>  </semantics>  <mi> n </mi>  </msub>  <mo> ( </mo>  <mrow>  <mi> w </mi>  <mo> + </mo>  <mi> z </mi>  </mrow>  <mo> ) </mo>  </mrow>  </mrow>  </mrow>  </mrow>  <annotation-xml encoding='MathML-Content'>  <apply>  <eq />  <apply>  <sum />  <bvar>  <ci> k </ci>  </bvar>  <lowlimit>  <cn type='integer'> 0 </cn>  </lowlimit>  <uplimit>  <ci> n </ci>  </uplimit>  <apply>  <times />  <apply>  <ci> Binomial </ci>  <ci> n </ci>  <ci> k </ci>  </apply>  <apply>  <ci> BernoulliB </ci>  <ci> k </ci>  <ci> z </ci>  </apply>  <apply>  <ci> BernoulliB </ci>  <apply>  <plus />  <ci> n </ci>  <apply>  <times />  <cn type='integer'> -1 </cn>  <ci> k </ci>  </apply>  </apply>  <ci> w </ci>  </apply>  </apply>  </apply>  <apply>  <plus />  <apply>  <times />  <ci> n </ci>  <apply>  <plus />  <ci> w </ci>  <ci> z </ci>  <cn type='integer'> -1 </cn>  </apply>  <apply>  <ci> BernoulliB </ci>  <apply>  <plus />  <ci> n </ci>  <cn type='integer'> -1 </cn>  </apply>  <apply>  <plus />  <ci> w </ci>  <ci> z </ci>  </apply>  </apply>  </apply>  <apply>  <times />  <cn type='integer'> -1 </cn>  <apply>  <times />  <apply>  <plus />  <ci> n </ci>  <cn type='integer'> -1 </cn>  </apply>  <apply>  <ci> BernoulliB </ci>  <ci> n </ci>  <apply>  <plus />  <ci> w </ci>  <ci> z </ci>  </apply>  </apply>  </apply>  </apply>  </apply>  </apply>  </annotation-xml>  </semantics>  </math> 
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 | | Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k_", "=", "0"]], "n_"], RowBox[List[RowBox[List["Binomial", "[", RowBox[List["n_", ",", "k_"]], "]"]], " ", RowBox[List["BernoulliB", "[", RowBox[List["k_", ",", "z_"]], "]"]], " ", RowBox[List["BernoulliB", "[", RowBox[List[RowBox[List["n_", "-", "k_"]], ",", "w_"]], "]"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["n", " ", RowBox[List["(", RowBox[List["z", "+", "w", "-", "1"]], ")"]], " ", RowBox[List["BernoulliB", "[", RowBox[List[RowBox[List["n", "-", "1"]], ",", RowBox[List["z", "+", "w"]]]], "]"]]]], "-", RowBox[List[RowBox[List["(", RowBox[List["n", "-", "1"]], ")"]], " ", RowBox[List["BernoulliB", "[", RowBox[List["n", ",", RowBox[List["z", "+", "w"]]]], "]"]]]]]]]]]] | 
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 | | Date Added to functions.wolfram.com (modification date) | 
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