StirlingS2[n, 6] == (1/120) (-1)^n (-1 + n)! (HarmonicNumber[-1 + n]^5 - 10 HarmonicNumber[-1 + n]^3 HarmonicNumber[-1 + n, 2] + 20 HarmonicNumber[-1 + n]^2 HarmonicNumber[-1 + n, 3] - 20 HarmonicNumber[-1 + n, 2] HarmonicNumber[-1 + n, 3] + 15 HarmonicNumber[-1 + n] (HarmonicNumber[-1 + n, 2]^2 - 2 HarmonicNumber[-1 + n, 4]) + 24 HarmonicNumber[-1 + n, 5]) /; Element[n, Integers] && n > 0

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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> <msubsup> <semantics> <mi> &#119982; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[ScriptCapitalS]&quot;, StirlingS2] </annotation> </semantics> <mi> n </mi> <mrow> <mo> ( </mo> <mn> 6 </mn> <mo> ) </mo> </mrow> </msubsup> <mo> &#10869; </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 120 </mn> </mfrac> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> n </mi> </msup> <mo> &#8290; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msubsup> <semantics> <mi> H </mi> <annotation-xml encoding='MathML-Content'> <ci> HarmonicNumber </ci> </annotation-xml> </semantics> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mn> 5 </mn> </msubsup> <mo> - </mo> <mrow> <mn> 10 </mn> <mo> &#8290; 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2007-05-02