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A272415 Asymptotic mean (normalized by n) of the third longest cycle in a random permutation on n symbols.

%I A272415 #6 Feb 16 2025 08:33:34
%S A272415 0,8,8,3,1,6,0,9,8,8,8,3,1,5,3,6,3,1,0,1,0,5,4,2,5,6,6,4,2,9,8,7,6,7,
%T A272415 0,1,1,7,2,3,6,4,3,2,0,4,5,1,1,6,3,3,3,0,4,6,6,7,8,7,4,0,9,3,0,9,4,2,
%U A272415 7,0,2,2,3,9,5,7,4,6,0,9,9,0,6,0,9,6,5,9,4,8,5,1,3,9,9,7,1,5,5
%N A272415 Asymptotic mean (normalized by n) of the third longest cycle in a random permutation on n symbols.
%D A272415 Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.4 Golomb-Dickman Constant, p. 285.
%H A272415 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 A272415 Eric Weisstein's MathWorld, <a href="https://mathworld.wolfram.com/Golomb-DickmanConstant.html">Golomb-Dickman Constant</a>
%H A272415 Wikipedia, <a href="http://en.wikipedia.org/wiki/Golomb%E2%80%93Dickman_constant">Golomb-Dickman constant</a>
%F A272415 Integral_{0..infinity} 1 - e^Ei(-x)*(1 - Ei(-x) + (1/2)*Ei(-x)^2) dx, where Ei is the exponential integral.
%e A272415 0.0883160988831536310105425664298767011723643204511633304667874093...
%t A272415 digits = 98; NIntegrate[1 - E^ExpIntegralEi[-x]*(1 - ExpIntegralEi[-x] + (1/2)*ExpIntegralEi[-x]^2), {x, 0, 100}, WorkingPrecision -> digits + 5] // Join[{0}, RealDigits[#, 10, digits][[1]]]&
%Y A272415 Cf. A084945, A247398, A272413, A272414.
%K A272415 nonn,cons
%O A272415 0,2
%A A272415 _Jean-François Alcover_, Apr 29 2016