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Denominators of coefficients in series expansion of Cl_2(x)+x*log(x), where Cl_2 is the Clausen function of order 2.
1

%I #13 Sep 08 2022 08:46:10

%S 1,1,1,72,1,14400,1,1270080,1,87091200,1,5269017600,1,203997201408000,

%T 1,15692092416000,1,2902409413263360000,1,1747310222272462848000,1,

%U 337200218333282304000000,1,7135156619932253552640000,1,1016294482039046201671680000000

%N Denominators of coefficients in series expansion of Cl_2(x)+x*log(x), where Cl_2 is the Clausen function of order 2.

%H Eric Weisstein's MathWorld, <a href="http://mathworld.wolfram.com/ClausenFunction.html">Clausen Function</a>

%H Eric Weisstein's MathWorld, <a href="http://mathworld.wolfram.com/ClausensIntegral.html">Clausen's Integral</a>

%F Denominators of BernoulliB(n - 1)/((n - 1)*n!), except the first 3 terms.

%e Coefficients begin 0, 1, 0, 1/72, 0, 1/14400, 0, 1/1270080, 0, 1/87091200, 0, 1/5269017600, 0, 691/203997201408000, ...

%t Clausen2[x_] := (I/2)*(PolyLog[2, Exp[-I*x]] - PolyLog[2, Exp[I*x]]); a[n_] := SeriesCoefficient[Clausen2[x] + x*Log[x], {x, 0, n}]; (* or *) a[n_] := If[Mod[n, 4] == 3, 1, -1]*BernoulliB[n - 1]/((n - 1)*n!); a[0] = a[2] = 0; a[1] = 1; Table[a[n] // Denominator, {n, 0, 30}] (* Apparently this only works with an older version of Mma *)

%t Flatten[{1, 1, Table[If[EvenQ[n], Denominator[Zeta[n]/(n*(n+1)*2^(n-1)*Pi^n)], 1],{n, 1, 20}]}] (* _Vaclav Kotesovec_, Nov 04 2014 *)

%o (Magma) [1,1,1] cat [Denominator(Bernoulli(n - 1)/((n - 1)*Factorial(n))) : n in [3..50]]; // _Vincenzo Librandi_, Nov 05 2014

%Y Cf. A027641, A027642, A249699.

%K nonn,frac

%O 0,4

%A _Jean-François Alcover_, Nov 04 2014