cp's OEIS Frontend

A274400 Decimal expansion of 'V', the value of a 4-dimensional iterated integral studied by David Broadhurst in connection with Quantum Field Theory (negated).

%I A274400 #9 Jun 20 2016 08:51:23
%S A274400 0,3,9,0,1,2,7,2,6,3,6,0,1,6,7,1,6,6,0,1,7,5,6,6,9,4,1,3,5,4,5,0,0,4,
%T A274400 6,1,7,5,7,4,7,5,8,5,7,1,3,8,6,1,3,0,9,9,0,1,4,9,3,8,9,6,7,3,9,5,4,0,
%U A274400 3,8,9,2,7,5,0,1,8,5,6,5,4,8,7,1,8,1,2,1,8,8,1,2,8,2,8,4,2,6,1,2,8,8
%N A274400 Decimal expansion of 'V', the value of a 4-dimensional iterated integral studied by David Broadhurst in connection with Quantum Field Theory (negated).
%D A274400 Jonathan Borwein and Peter Borwein, Experimental and Computational Mathematics: Selected Writings, Perfectly Scientific Press, 2010, p. 106.
%H A274400 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;
%F A274400 V = Sum_{j>k>0} (-1)^j cos(2Pi k/3)/(j^3 k).
%F A274400 Equals 3 zeta(3)/8-1/2+Sum_{k>=2} ((-1)^k*((24*(k - 1)*(3*k - 4))/(3*k - 2)^3 + (8*(3*k*(3*k - 5) + 4))/(27*(k - 1)^3) + PolyGamma(2, (3*k)/2 - 1) - PolyGamma(2, (3*(k - 1))/2)))/(48*(k - 1)*(3*k - 4)*(3*k - 2)).
%e A274400 -0.0390127263601671660175669413545004617574758571386130990149389673954...
%t A274400 digits = 101;
%t A274400 v[k_] := ((-1)^k*((24*(k - 1)*(3*k - 4))/(3*k - 2)^3 + (8*(3*k*(3*k - 5) + 4))/(27*(k - 1)^3) + PolyGamma[2, (3*k)/2 - 1] - PolyGamma[2, (3*(k - 1))/2]))/(48*(k - 1)*(3*k - 4)*(3*k - 2));
%t A274400 V = 3 Zeta[3]/8 - 1/2 + NSum[v[k], {k, 2, Infinity}, WorkingPrecision -> digits + 10, Method -> "AlternatingSigns"]; Join[{0},
%t A274400 RealDigits[V, 10, digits][[1]]]
%Y A274400 Cf. A255685 ('U' in Borwein & Borwein).
%K A274400 nonn,cons
%O A274400 0,2
%A A274400 _Jean-François Alcover_, Jun 20 2016