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A272427 Asymptotic variance (normalized by n^2) of the third longest cycle in a random permutation on n symbols.

%I A272427 #10 Feb 16 2025 08:33:34
%S A272427 0,0,4,4,9,3,9,2,3,1,8,1,7,9,0,8,0,4,7,4,7,9,4,4,9,2,2,0,5,7,5,6,9,9,
%T A272427 6,9,2,6,4,9,3,1,9,7,8,4,3,0,7,7,0,7,2,4,2,0,7,5,0,5,9,2,3,9,8,0,0,3,
%U A272427 5,0,0,7,5,4,0,9,8,6,0,4,8,4,2,8,1,9,3,8,7,5,8,6,9,5,9,3,0,1,8,0
%N A272427 Asymptotic variance (normalized by n^2) of the third longest cycle in a random permutation on n symbols.
%D A272427 Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.4 Golomb-Dickman Constant, p. 285.
%H A272427 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 A272427 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Golomb-DickmanConstant.html">Golomb-Dickman Constant</a>
%H A272427 Wikipedia, <a href="http://en.wikipedia.org/wiki/Golomb%E2%80%93Dickman_constant">Golomb-Dickman constant</a>
%F A272427 Integral_{0..infinity} x*(1 - e^Ei(-x)*(1 - Ei(-x) + (1/2)*Ei(-x)^2)) dx - (Integral_{0..infinity} 1 - e^Ei(-x)*(1 - Ei(-x) + (1/2)*Ei(-x)^2) dx)^2, where Ei is the exponential integral.
%e A272427 0.00449392318179080474794492205756996926493197843077072420750592398...
%t A272427 digits = 98; Ei = ExpIntegralEi; NIntegrate[x*(1 - E^Ei[-x]*(1 - Ei[-x] + (1/2)*Ei[-x]^2)), {x, 0, 100}, WorkingPrecision -> digits + 5] - NIntegrate[1 - E^Ei[-x]*(1 - Ei[-x] + (1/2)*Ei[-x]^2), {x, 0, 100}, WorkingPrecision -> digits + 5]^2 // Join[{0, 0}, RealDigits[#, 10, digits][[1]]]&
%Y A272427 Cf. A084945, A247398, A272413, A272414, A272415.
%K A272427 nonn,cons
%O A272427 0,3
%A A272427 _Jean-François Alcover_, Apr 29 2016