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

%I #24 Dec 05 2018 08:10:46

%S 0,1,0,1,0,1,0,1,0,1,0,1,0,691,0,1,0,3617,0,43867,0,174611,0,77683,0,

%T 236364091,0,657931,0,3392780147,0,1723168255201,0,7709321041217,0,

%U 151628697551,0,26315271553053477373,0,154210205991661,0,261082718496449122051,0

%N Numerators 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 Numerators 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, ...

%p A249699List := proc(len) local mu, ser;

%p mu := h -> sum(bernoulli(2*k)/(2*k)!*h^(2*k-1), k=0..infinity);

%p ser := series(mu(h), h, len+2): seq((-1)^binomial(n,2)*numer(coeff(ser, h, n)), n=0..len): 0, 1, op([%]) end: A249699List(48); # _Peter Luschny_, Dec 05 2018

%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] // Numerator, {n, 0, 30}] (* Apparently this only works with an older version of Mma *)

%t Flatten[{0, 1, Table[If[EvenQ[n], Numerator[Zeta[n]/(n*(n+1)*2^(n-1)*Pi^n)], 0], {n, 1, 30}]}] (* _Vaclav Kotesovec_, Nov 04 2014 *)

%Y Cf. A027641, A027642, A249700.

%K nonn,frac

%O 0,14

%A _Jean-François Alcover_, Nov 04 2014

%E More terms from _Peter Luschny_, Dec 05 2018