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Asymptotic mean (normalized by n) of the second longest cycle in a random permutation on n symbols.
3

%I #5 Apr 29 2016 10:16:13

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

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

%U 3,1,0,8,8,4,6,4,2,8,7,2,6,3,4,2,8,7,1,6,3,6,8,2,9,8,8,3,4,7

%N Asymptotic mean (normalized by n) of the second longest cycle in a random permutation on n symbols.

%D Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.4 Golomb-Dickman Constant, p. 285.

%H Xavier Gourdon, <a href="http://algo.inria.fr/gourdon/thesis.html">Combinatoire, Algorithmique et Géométrie des Polynomes</a> Ecole Polytechnique, Paris 1996, page 152 [in French]

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

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Golomb%E2%80%93Dickman_constant">Golomb-Dickman constant</a>

%F Integral_{0..infinity} 1 - exp(Ei(-x))*(1 - Ei(-x)) dx, where Ei is the exponential integral.

%e 0.20958087428418581398902965781534955690113103201623433...

%t digits = 98; NIntegrate[1 - Exp[ExpIntegralEi[-x]]*(1 - ExpIntegralEi[-x]), {x, 0, 200}, WorkingPrecision -> digits+5] // RealDigits[#, 10, digits]& // First

%Y Cf. A084945, A247398.

%K nonn,cons

%O 0,1

%A _Jean-François Alcover_, Apr 29 2016