Sum[PartitionsP[k] t^k, {k, 0, Infinity}] == Product[1/(1 - t^k), {k, 1, Infinity}]

Cell[BoxData[RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[RowBox[List["PartitionsP", "[", "k", "]"]], SuperscriptBox["t", "k"]]]]], "\[Equal]", " ", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "1"]], "\[Infinity]"], FractionBox["1", RowBox[List["1", "-", SuperscriptBox["t", "k"]]]]]]]]]]

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mrow> <mrow> <semantics> <mi> p </mi> <annotation encoding='Mathematica'> TagBox[&quot;p&quot;, PartitionsP] </annotation> </semantics> <mo> ( </mo> <mi> k </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> t </mi> <mi> k </mi> </msup> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <munderover> <mo> &#8719; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mfrac> <mn> 1 </mn> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> t </mi> <mi> k </mi> </msup> </mrow> </mfrac> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <ci> PartitionsP </ci> <ci> k </ci> </apply> <apply> <power /> <ci> t </ci> <ci> k </ci> </apply> </apply> </apply> <apply> <product /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> t </ci> <ci> k </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k_", "=", "0"]], "\[Infinity]"], RowBox[List[RowBox[List["PartitionsP", "[", "k_", "]"]], " ", SuperscriptBox["t_", "k_"]]]]], "]"]], "\[RuleDelayed]", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "1"]], "\[Infinity]"], FractionBox["1", RowBox[List["1", "-", SuperscriptBox["t", "k"]]]]]]]]]]

2001-10-29