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%I #14 Oct 01 2018 03:32:09
%S 2,6,0,4,3,4,1,3,7,6,3,2,1,6,2,0,9,8,9,5,5,7,2,9,1,4,3,2,0,8,0,3,0,7,
%T 8,5,4,5,5,0,4,4,7,7,8,8,4,8,4,2,8,4,7,3,4,0,7,3,6,6,6,8,7,6,5,5,6,2,
%U 8,9,9,4,8,8,3,8,7,2,7,3,9,3,6,4,2,8,9,8,5,6,9,4,4,0,6,9,9,5,3,6,7,3,6,8
%N Decimal expansion of S_2 = Sum_{n>=0} (2n+1)/((3n+1)^2 (3n+2)^2), a constant related to Quantum Field Theory (see the paper by David Broadhurst).
%H D. J. Broadhurst, <a href="http://arxiv.org/abs/hep-th/9803091">Massive 3-loop Feynman diagrams reducible to SC* primitives of algebras of the sixth root of unity</a>, arXiv:hep-th/9803091, 1998;
%H Eric Weisstein's MathWorld, <a href="http://mathworld.wolfram.com/PolygammaFunction.html">Polygamma Function</a>.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Polygamma_function">Polygamma Function</a>.
%F S_2 = (1/27)*(PolyGamma(1, 1/3) - PolyGamma(1, 2/3)).
%F Also equals 2/3^(3/2) Cl_2(2Pi/3) where Cl_2 is the Clausen function Cl_2(x) = Sum_{n>0} sin(n x)/n^2.
%e 0.2604341376321620989557291432080307854550447788484284734073666876556...
%t S2 = (1/27)*(PolyGamma[1, 1/3] - PolyGamma[1, 2/3]);
%t RealDigits[S2, 10, 104][[1]]
%o (PARI)
%o polygamma(n, x) = if (n == 0, psi(x), (-1)^(n+1)*n!*zetahurwitz(n+1, x));
%o (polygamma(1, 1/3) - polygamma(1, 2/3))/27 \\ _Gheorghe Coserea_, Sep 30 2018
%o (PARI)
%o clausen(n, x) = my(z = polylog(n, exp(I*x))); if (n%2, real(z), imag(z));
%o 2/3^(3/2) * clausen(2, 2*Pi/3) \\ _Gheorghe Coserea_, Sep 30 2018
%o (PARI)
%o sumpos(n=0, (2*n+1)/((3*n+1)^2*(3*n+2)^2)) \\ _Gheorghe Coserea_, Sep 30 2018
%o (PARI)
%o 4/81*sumalt(n=0, (-1/27)^n*(9/(6*n+1)^2 - 9/(6*n+2)^2 - 12/(6*n+3)^2 - 3/(6*n+4)^2 + 1/(6*n+5)^2)) \\ _Gheorghe Coserea_, Sep 30 2018
%K nonn,cons
%O 0,1
%A _Jean-François Alcover_, Jun 20 2016
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