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Decimal expansion of 1-delta_0, where delta_0 is the Hall-Montgomery constant (A143301).
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%I #23 Apr 22 2021 04:46:22

%S 8,2,8,4,9,9,5,0,6,8,5,8,4,6,3,9,3,4,1,3,9,5,6,0,0,2,8,4,4,4,7,8,7,8,

%T 9,0,3,7,7,7,3,7,0,9,5,7,7,0,4,4,9,1,5,8,2,8,5,7,8,8,9,0,8,1,7,6,3,0,

%U 1,3,9,4,4,0,5,6,9,1,4,2,2,0,1,2,0,2,8,8,0,1,9,1,3,1,9,9,1,8,2,6,9

%N Decimal expansion of 1-delta_0, where delta_0 is the Hall-Montgomery constant (A143301).

%C This constant, by coincidence, is also a limiting probability concerning the number of cycles of a given length in a random permutation.

%C One has P_1(xi) = 1-delta_0 = Pi^2/6 - log(xi) - log(xi)^2 - 2*Li_2(xi), where xi = 1/(1+sqrt(e)) (see A246848 and the references).

%H Steven R. Finch, <a href="http://arxiv.org/abs/2001.00578">Errata and Addenda to Mathematical Constants</a>, arXiv:2001.00578 [math.HO], 2020, p. 29.

%H Michael Lugo, <a href="http://arxiv.org/abs/0909.2909">The number of cycles of specified normalized length in permutations</a>, arXiv:0909.2909 [math.CO], 2009.

%H Eric Weisstein's MathWorld, <a href="http://mathworld.wolfram.com/Hall-MontgomeryConstant.html">Hall-Montgomery Constant</a>

%F Pi^2/6 + log(1 + sqrt(e)) - log(1 + sqrt(e))^2 - 2*Li_2(1/(1 + sqrt(e))), where Li_2 is the dilogarithm function.

%e 0.82849950685846393413956002844478789037773709577...

%t Pi^2/6 + Log[1 + Sqrt[E]] - Log[1 + Sqrt[E]]^2 - 2*PolyLog[2, 1/(1 + Sqrt[E])] // RealDigits[#, 10, 101]& // First

%o (PARI) Pi^2/6 + log(exp(1/2)+1) - log(exp(1/2)+1)^2 - 2*polylog(2, 1/(exp(1/2)+1)) \\ _Charles R Greathouse IV_, Sep 08 2014

%o (Python)

%o from mpmath import mp, log, exp, polylog, pi

%o mp.dps=102

%o print([int(n) for n in list(str(pi**2/6 + log(exp(1/2)+1) - log(exp(1/2)+1)**2 - 2*polylog(2, 1/(exp(1/2)+1)))[2:-1])]) # _Indranil Ghosh_, Jul 04 2017

%Y Cf. A143301, A246848.

%K nonn,cons

%O 0,1

%A _Jean-François Alcover_, Sep 05 2014