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A274439
Decimal expansion of Q(1), value of one of five integrals related to Quantum Field Theory (see the paper by David Broadhurst).
5
2, 6, 3, 6, 1, 8, 5, 7, 2, 5, 2, 2, 4, 8, 7, 2, 2, 2, 6, 5, 4, 6, 4, 0, 2, 0, 4, 7, 9, 1, 9, 8, 6, 8, 6, 8, 5, 5, 3, 3, 9, 5, 2, 4, 3, 7, 4, 0, 8, 5, 4, 6, 5, 0, 4, 9, 6, 2, 6, 1, 4, 3, 4, 0, 2, 7, 6, 6, 5, 5, 4, 3, 8, 2, 5, 1, 8, 2, 0, 4, 0, 7, 9, 4, 7, 0, 6, 6, 7, 0, 6, 1, 6, 0, 6, 2, 2, 0, 5, 4, 7, 6, 6
OFFSET
1,1
FORMULA
Q(n) = Integral_{x>0} arccosh((x+2)/2)^2 log((x+1)/x)/(x+n) dx.
Computation is done using the analytical form given by David Broadhurst:
Q(1) = (4/3)*Cl2(Pi/3)^2 + (7/6)*zeta(4), where Cl_2 is the Clausen integral.
EXAMPLE
2.636185725224872226546402047919868685533952437408546504962614340...
MATHEMATICA
Cl2[x_] := (I/2)*(PolyLog[2, Exp[-I*x]] - PolyLog[2, Exp[I*x]]);
Q[1] = 4/3 Cl2[Pi/3]^2 + 7/6 Zeta[4];
RealDigits[N[Q[1], 103] // Chop][[1]]
PROG
(PARI)
Q(n) = intnum(x=0, oo, acosh((x+2)/2)^2 * log((x+1)/x)/(x+n));
Q(1) \\ Gheorghe Coserea, Sep 30 2018
(PARI)
clausen(n, x) = my(z = polylog(n, exp(I*x))); if (n%2, real(z), imag(z));
4/3*clausen(2, Pi/3)^2 + 7/6*zeta(4) \\ Gheorghe Coserea, Sep 30 2018
CROSSREFS
Cf. A274439 (Q(1)), A274440 (Q(2)), A274441 (Q(3)), A274442 (Q(4)).
Sequence in context: A139384 A083481 A177209 * A280342 A275476 A185380
KEYWORD
nonn,cons
AUTHOR
STATUS
approved