A249699 Numerators of coefficients in series expansion of Cl_2(x)+x*log(x), where Cl_2 is the Clausen function of order 2.

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, 236364091, 0, 657931, 0, 3392780147, 0, 1723168255201, 0, 7709321041217, 0, 151628697551, 0, 26315271553053477373, 0, 154210205991661, 0, 261082718496449122051, 0

Offset

0,14

Links

Table of n, a(n) for n=0..42.

Eric Weisstein's MathWorld, Clausen Function

Eric Weisstein's MathWorld, Clausen's Integral

Formula

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

Example

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

Maple

A249699List := proc(len) local mu, ser;
mu  := h -> sum(bernoulli(2*k)/(2*k)!*h^(2*k-1), k=0..infinity);
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

Mathematica

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 *)
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 *)

Crossrefs

Cf. A027641, A027642, A249700.

Sequence in context: A120084 A120082 A358625 * A141588 A281331 A281332

Adjacent sequences: A249696 A249697 A249698 * A249700 A249701 A249702

Keyword

nonn,frac

Author

Jean-François Alcover, Nov 04 2014

Extensions

More terms from Peter Luschny, Dec 05 2018

Status

approved