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

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, 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, 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

Offset

0,2

References

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

Links

Table of n, a(n) for n=0..101.

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;

Formula

V = Sum_{j>k>0} (-1)^j cos(2Pi k/3)/(j^3 k).

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)).

Example

-0.0390127263601671660175669413545004617574758571386130990149389673954...

Mathematica

digits = 101;
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));
V = 3 Zeta[3]/8 - 1/2 + NSum[v[k], {k, 2, Infinity}, WorkingPrecision -> digits + 10, Method -> "AlternatingSigns"]; Join[{0},
RealDigits[V, 10, digits][[1]]]

Crossrefs

Cf. A255685 ('U' in Borwein & Borwein).

Sequence in context: A088110 A122759 A247042 * A200495 A329937 A272535

Adjacent sequences: A274397 A274398 A274399 * A274401 A274402 A274403

Keyword

nonn,cons

Author

Jean-François Alcover, Jun 20 2016

Status

approved