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%I #19 Jan 17 2020 16:13:37
%S 1,0,7,2,9,9,7,9,4,4,3,8,9,5,2,7,0,1,7,7,3,7,9,7,1,3,9,4,9,5,4,4,6,5,
%T 5,5,5,6,8,1,0,1,4,1,8,8,3,1,4,3,7,4,0,4,5,6,6,7,8,5,3,5,2,4,3,1,9,7,
%U 8,4,5,0,2,2,4,8,6,2,7,2,8,1,1,9,6,2,1,9,2,6,4,1,1,9,5,7,0,3,4,1,5,5,9,2,7
%N Decimal expansion of the coefficient c appearing in the asymptotic expression of the probability that a random n-permutation is a cube as c/n^3.
%H Steven R. Finch, <a href="http://arxiv.org/abs/2001.00578">Errata and Addenda to Mathematical Constants</a>, p. 36.
%F c = 3^(5/6)*Gamma(1/3)/(2*Pi)*prod_{k>=1} psi(1/(3k)), where psi(x) = 1/3*(e^x + 2*e^(-x/2)*cos(sqrt(3)*(x/2))).
%e 1.072997944389527017737971394954465555681...
%p evalf(3^(5/6) * GAMMA(1/3) / (2*Pi) * Product(1/3*(exp(1/(3*k)) + 2*exp(-1/(6*k)) * cos(sqrt(3)/(6*k))), k=1..infinity),100) # _Vaclav Kotesovec_, Sep 17 2014
%t digits = 40; m0 = 1000; dm = 1000; psi[x_] := 1/3*(E^x + 2*E^(-x/2)*Cos[Sqrt[3]*(x/2)]); tail[m_] := (-98761420800*PolyGamma[2, m] - 4572288*PolyGamma[5, m] - 53*PolyGamma[8, m])/31998700339200; Clear[f]; f[m_] := f[m] = Sum[Log[psi[1/(3*k)]], {k, 1, m - 1}] + tail[m] // N[#, digits + 10] &; f[m0]; f[m = m0 + dm]; While[RealDigits[f[m], 10, digits + 5] != RealDigits[f[m - dm], 10, digits + 5], Print["f(", m, ") = ", f[m]]; m = m + dm]; c = 3^(5/6)*Gamma[1/3]/(2*Pi)*E^f[m]; RealDigits[c, 10, 40] // First
%o (PARI) default(realprecision,150); 3^(5/6) * gamma(1/3) / (2*Pi) * exp(sumpos(k=1,log(1/3*(exp(1/(3*k)) + 2*exp(-1/(6*k)) * cos(sqrt(3)/(6*k)))))) \\ _Vaclav Kotesovec_, Sep 21 2014
%Y Cf. A246945.
%K nonn,cons
%O 1,3
%A _Jean-François Alcover_, Sep 08 2014
%E More terms from _Vaclav Kotesovec_, Sep 17 2014
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