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http://functions.wolfram.com/04.16.17.0002.01
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PartitionsP[n] == Sum[(-1)^(k - 1) (PartitionsP[n - (3 k^2 - k)/2] +
PartitionsP[n - (3 k^2 + k)/2]), {k, 1, n}]
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Cell[BoxData[RowBox[List[RowBox[List["PartitionsP", "[", "n", "]"]], "\[Equal]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "n"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["k", "-", "1"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["PartitionsP", "[", RowBox[List["n", "-", FractionBox[RowBox[List[RowBox[List["3", " ", SuperscriptBox["k", "2"]]], "-", "k"]], "2"]]], " ", "]"]], "+", RowBox[List["PartitionsP", "[", RowBox[List["n", "-", FractionBox[RowBox[List[RowBox[List["3", " ", SuperscriptBox["k", "2"]]], "+", "k"]], "2"]]], " ", "]"]]]], ")"]]]]]]]]]]
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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> <semantics> <mi> p </mi> <annotation encoding='Mathematica'> TagBox["p", PartitionsP] </annotation> </semantics> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </munderover> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> k </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <semantics> <mi> p </mi> <annotation encoding='Mathematica'> TagBox["p", PartitionsP] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> n </mi> <mo> - </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 3 </mn> <mo> ⁢ </mo> <msup> <mi> k </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mi> k </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <semantics> <mi> p </mi> <annotation encoding='Mathematica'> TagBox["p", PartitionsP] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> n </mi> <mo> - </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 3 </mn> <mo> ⁢ </mo> <msup> <mi> k </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mi> k </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> PartitionsP </ci> <ci> n </ci> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> n </ci> </uplimit> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <apply> <plus /> <ci> k </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <ci> PartitionsP </ci> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <power /> <ci> k </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <ci> PartitionsP </ci> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <power /> <ci> k </ci> <cn type='integer'> 2 </cn> </apply> </apply> <ci> k </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["PartitionsP", "[", "n_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "n"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["k", "-", "1"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["PartitionsP", "[", RowBox[List["n", "-", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["3", " ", SuperscriptBox["k", "2"]]], "-", "k"]], ")"]]]]]], "]"]], "+", RowBox[List["PartitionsP", "[", RowBox[List["n", "-", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["3", " ", SuperscriptBox["k", "2"]]], "+", "k"]], ")"]]]]]], "]"]]]], ")"]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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