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A274441
Decimal expansion of Q(3), value of one of five integrals related to Quantum Field Theory (see the paper by David Broadhurst).
4
2, 0, 3, 4, 3, 6, 8, 9, 7, 1, 3, 1, 7, 2, 0, 4, 4, 4, 7, 1, 5, 4, 1, 0, 0, 4, 8, 2, 3, 2, 7, 0, 6, 9, 9, 8, 1, 9, 7, 6, 9, 5, 0, 4, 7, 3, 6, 5, 1, 2, 8, 6, 4, 5, 7, 0, 8, 4, 4, 3, 7, 2, 3, 9, 3, 8, 0, 6, 5, 7, 3, 4, 1, 9, 6, 4, 9, 6, 6, 2, 4, 5, 6, 2, 2, 3, 9, 0, 3, 6, 7, 8, 3, 6, 5, 5, 0, 1, 4, 2, 5
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(3) = -50/9 Cl2(Pi/3)^2+596/81 zeta(4)-16/9 U+32/3 V, where Cl_2 is the Clausen integral, U A255685 and V A274400.
EXAMPLE
2.03436897131720444715410048232706998197695047365128645708443723938...
MATHEMATICA
digits = 101;
Cl2[x_] := (I/2)*(PolyLog[2, Exp[-I*x]] - PolyLog[2, Exp[I*x]]);
U = A255685 = Pi^4/180 + (Pi^2/12)*Log[2]^2 - (1/12)*Log[2]^4 - 2*PolyLog[4, 1/2];
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 = A274400 = 3 Zeta[3]/8 - 1/2 + NSum[v[k], {k, 2, Infinity}, WorkingPrecision -> digits + 10, Method -> "AlternatingSigns"];
Q[3] = -50/9 Cl2[Pi/3]^2 + 596/81 Zeta[4] - 16/9 U + 32/3 V;
RealDigits[N[Q[3], digits] // Chop][[1]]
PROG
(PARI)
Q(n) = intnum(x=0, oo, acosh((x+2)/2)^2 * log((x+1)/x)/(x+n));
Q(3) \\ Gheorghe Coserea, Sep 30 2018
(PARI)
clausen(n, x) = my(z = polylog(n, exp(I*x))); if (n%2, real(z), imag(z));
polygamma(n, x) = if (n == 0, psi(x), (-1)^(n+1)*n!*zetahurwitz(n+1, x));
u31=Pi^4/180 + (Pi^2/12)*log(2)^2 - (1/12)*log(2)^4 - 2*polylog(4, 1/2);
v31=3*zeta(3)/8 - 1/2 + sumalt(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)));
-50/9*clausen(2, Pi/3)^2 + 596/81*zeta(4) - 16/9*u31 + 32/3*v31 \\ Gheorghe Coserea, Sep 30 2018
CROSSREFS
Cf. A274438 (Q(0)), A274439 (Q(1)), A274440 (Q(2)), A274442 (Q(4)).
Sequence in context: A287016 A368312 A285722 * A213859 A330492 A350534
KEYWORD
nonn,cons
AUTHOR
STATUS
approved