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 A274402 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). 0
 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, 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, 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 (list; constant; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 LINKS D. J. Broadhurst, Massive 3-loop Feynman diagrams reducible to SC* primitives of algebras of the sixth root of unity, arXiv:hep-th/9803091, 1998; Eric Weisstein's MathWorld, Polygamma Function. Wikipedia, Polygamma Function. FORMULA S_2 = (1/27)*(PolyGamma(1, 1/3) - PolyGamma(1, 2/3)). 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. EXAMPLE 0.2604341376321620989557291432080307854550447788484284734073666876556... MATHEMATICA S2 = (1/27)*(PolyGamma[1, 1/3] - PolyGamma[1, 2/3]); RealDigits[S2, 10, 104][[1]] PROG (PARI) polygamma(n, x) = if (n == 0, psi(x), (-1)^(n+1)*n!*zetahurwitz(n+1, x)); (polygamma(1, 1/3) - polygamma(1, 2/3))/27 \\ Gheorghe Coserea, Sep 30 2018 (PARI) clausen(n, x) = my(z = polylog(n, exp(I*x))); if (n%2, real(z), imag(z)); 2/3^(3/2) * clausen(2, 2*Pi/3) \\ Gheorghe Coserea, Sep 30 2018 (PARI) sumpos(n=0, (2*n+1)/((3*n+1)^2*(3*n+2)^2)) \\ Gheorghe Coserea, Sep 30 2018 (PARI) 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 CROSSREFS Sequence in context: A084897 A021388 A011040 * A115252 A108431 A190144 Adjacent sequences:  A274399 A274400 A274401 * A274403 A274404 A274405 KEYWORD nonn,cons AUTHOR Jean-François Alcover, Jun 20 2016 STATUS approved

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Last modified February 19 18:30 EST 2019. Contains 320327 sequences. (Running on oeis4.)