CatalanNumber[z] \[Proportional] ((-1)^n 2^(-1 - 2 n))/(Sqrt[Pi] Gamma[3/2 - n] n! (z + 1/2 + n)) + ((-1)^n 2^(-1 - 2 n) (Log[4] - PolyGamma[3/2 - n] + PolyGamma[1 + n]))/ (Sqrt[Pi] Gamma[3/2 - n] n!) + (((-1)^n 4^(-1 - n))/(3 Sqrt[Pi] Gamma[3/2 - n] n!)) (Pi^2 + 3 Log[4]^2 + 3 PolyGamma[3/2 - n]^2 - 6 PolyGamma[3/2 - n] (Log[4] + PolyGamma[1 + n]) + 3 PolyGamma[1 + n] (Log[16] + PolyGamma[1 + n]) - 3 PolyGamma[1, 3/2 - n] - 3 PolyGamma[1, 1 + n]) (z + 1/2 + n) + (((-1)^n 4^(-1 - n))/(3 Sqrt[Pi] Gamma[3/2 - n] n!)) (Pi^2 Log[4] + Log[4]^3 - PolyGamma[3/2 - n]^3 + Log[64] PolyGamma[1 + n]^2 + PolyGamma[1 + n]^3 + 3 PolyGamma[3/2 - n]^2 (Log[4] + PolyGamma[1 + n]) + PolyGamma[1 + n] (Pi^2 + 3 Log[4]^2 - 3 PolyGamma[1, 3/2 - n] - 3 PolyGamma[1, 1 + n]) - PolyGamma[3/2 - n] (Pi^2 + 3 Log[4]^2 + 3 PolyGamma[1 + n] (Log[16] + PolyGamma[1 + n]) - 3 PolyGamma[1, 3/2 - n] - 3 PolyGamma[1, 1 + n]) - 3 Log[4] (PolyGamma[1, 3/2 - n] + PolyGamma[1, 1 + n]) - PolyGamma[2, 3/2 - n] + PolyGamma[2, 1 + n]) (z + 1/2 + n)^2 + \[Ellipsis] /; (z -> -(1/2) - n) && Element[n, Integers] && n >= 0

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&#8289; </mo> <mo> ( </mo> <mn> 4 </mn> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mn> 16 </mn> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mrow> <msup> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mrow> <mo> ( </mo> <mn> 1 </mn> <mo> ) </mo> </mrow> </msup> <mo> ( </mo> <mrow> <mfrac> <mn> 3 </mn> <mn> 2 </mn> </mfrac> <mo> - </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mrow> <msup> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mrow> <mo> ( </mo> <mn> 1 </mn> <mo> ) </mo> </mrow> </msup> <mo> ( </mo> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> + </mo> <mi> z </mi> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> n </mi> </msup> <mo> &#8290; </mo> <msup> <mn> 4 </mn> <mrow> <mrow> <mo> - </mo> <mi> n </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> </mrow> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <msqrt> <mi> &#960; </mi> </msqrt> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mfrac> <mn> 3 </mn> <mn> 2 </mn> </mfrac> <mo> - </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> n </mi> <mo> ! </mo> </mrow> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mi> log </mi> <mn> 3 </mn> </msup> <mo> ( </mo> <mn> 4 </mn> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <msup> <mi> &#960; </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mn> 4 </mn> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> 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2007-05-02