A274420 - OEIS

%I #6 Jun 21 2016 10:13:03

%S 8,2,1,6,8,5,9,8,1,7,5,0,8,7,3,8,0,6,2,9,1,3,3,9,8,3,3,8,6,0,1,0,8,5,

%T 8,2,4,9,6,9,5,0,8,3,3,9,1,7,2,5,7,5,0,3,6,8,3,5,5,7,5,7,9,1,1,5,3,3,

%U 5,1,9,6,8,1,6,3,1,9,2,6,2,3,1,2,2,4,2,9,9,0,3,4,1,4,0,6,1,1,9,6,8

%N Decimal expansion of V_5, a Quantum Field Theory constant [negated] related to the coloring of the tetrahedron with five masses.

%D Jonathan Borwein and Peter Borwein, Experimental and Computational Mathematics: Selected Writings, Perfectly Scientific Press, 2010, p. 106.

%H David 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; p. 18.

%F V_5 = 6 zeta(3) - 469/27 zeta(4) + 8/3 C^2 - 16 V, where C is A143298 and V A274400.

%e -8.21685981750873806291339833860108582496950833917257503683557579115...

%t digits = 101;

%t C0 = A143298 = (9 - PolyGamma[1, 2/3] + PolyGamma[1, 4/3])/(4*Sqrt[3]);

%t 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 V = A274400 = 3 Zeta[3]/8 - 1/2 + NSum[v[k], {k, 2, Infinity}, WorkingPrecision -> digits + 10, Method -> "AlternatingSigns"];

%t V5 = 6 Zeta[3] - 469/27 Zeta[4] + 8/3 C0^2 - 16 V;

%t RealDigits[V5, 10, digits][[1]]

%Y Cf. A274412 (V_1), A274413 (V_2A), A274414 (V_2N), A274415 (V_3T), A274416 (V_3S), A274417 (V_3L), A274418 (V_4A), A274419 (V_4N), A274421 (V_6).

%K nonn,cons

%O 1,1

%A _Jean-François Alcover_, Jun 21 2016