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A274438
Decimal expansion of Q(0), value of one of five integrals related to Quantum Field Theory (see the paper by David Broadhurst).
3
4, 1, 2, 0, 4, 2, 5, 8, 5, 7, 6, 8, 5, 6, 3, 3, 0, 0, 9, 3, 3, 3, 1, 9, 3, 2, 0, 5, 8, 6, 5, 5, 1, 8, 3, 9, 6, 8, 9, 0, 2, 2, 8, 9, 8, 0, 5, 1, 0, 0, 9, 5, 3, 3, 7, 9, 9, 7, 4, 2, 6, 2, 6, 6, 7, 7, 5, 5, 4, 4, 1, 5, 8, 1, 0, 1, 0, 7, 0, 2, 6, 0, 8, 9, 2, 0, 1, 6, 3, 9, 2, 6, 8, 5, 9, 1, 6, 4, 5, 3, 9, 8, 2, 9
OFFSET
1,1
FORMULA
Q(n) = Integral_{0..inf} arccosh((x+2)/2)^2 log((x+1)/x)/(x+n) dx.
Computation is done using the analytical form given by David Broadhurst: Q(0) = 4 Cl_2(Pi/3)^2, where Cl_2 is the Clausen integral.
15 Q(0) + 144 Q(1) - 448 Q(2) + 126 Q(3) + 168 Q(4) = 0.
EXAMPLE
4.1204258576856330093331932058655183968902289805100953379974262667755...
MATHEMATICA
Cl2[x_] := (I/2)*(PolyLog[2, Exp[-I*x]] - PolyLog[2, Exp[I*x]]);
Q[0] = 4 Cl2[Pi/3]^2 ;
RealDigits[N[Q[0], 104] // Chop][[1]]
PROG
(PARI)
Q(n) = intnum(x=0, oo, acosh((x+2)/2)^2 * log((x+1)/x)/(x+n));
Q(0) \\ Gheorghe Coserea, Oct 01 2018
(PARI)
clausen(n, x) = my(z = polylog(n, exp(I*x))); if (n%2, real(z), imag(z));
4*clausen(2, Pi/3)^2 \\ Gheorghe Coserea, Oct 01 2018
CROSSREFS
Cf. A274439 (Q(1)), A274440 (Q(2)), A274441 (Q(3)), A274442 (Q(4)).
Sequence in context: A343635 A287647 A331749 * A365387 A087565 A079163
KEYWORD
nonn,cons
AUTHOR
STATUS
approved