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A272414
Asymptotic variance (normalized by n^2) of the second longest cycle in a random permutation on n symbols.
2
0, 1, 2, 5, 5, 3, 7, 9, 0, 6, 3, 5, 9, 0, 5, 8, 7, 8, 1, 4, 7, 9, 8, 0, 0, 3, 5, 8, 4, 6, 6, 0, 1, 9, 8, 6, 7, 8, 5, 5, 0, 8, 3, 0, 1, 1, 9, 9, 3, 6, 5, 1, 7, 7, 2, 5, 9, 2, 4, 2, 5, 4, 2, 6, 7, 3, 9, 4, 6, 4, 9, 1, 4, 5, 7, 4, 3, 9, 7, 4, 9, 4, 2, 8, 8, 7, 3, 5, 1, 6, 5, 9, 3, 6, 2, 3, 5, 6, 6
OFFSET
0,3
REFERENCES
Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.4 Golomb-Dickman Constant, p. 285.
LINKS
Xavier Gourdon, Combinatoire, Algorithmique et Géométrie des Polynomes Ecole Polytechnique, Paris 1996, page 152 [in French]
Eric Weisstein's MathWorld, Golomb-Dickman Constant
FORMULA
Integral_{0..infinity} x*(1 - exp(Ei(-x))*(1 - Ei(-x))) dx - (integral_{0..infinity} 1 - exp(Ei(-x))*(1 - Ei(-x)) dx)^2, where Ei is the exponential integral.
EXAMPLE
0.012553790635905878147980035846601986785508301199365...
MATHEMATICA
digits = 98; NIntegrate[x*(1 - E^ExpIntegralEi[-x]*(1 - ExpIntegralEi[-x]) ), {x, 0, 200}, WorkingPrecision -> digits + 5] - NIntegrate[1 - E^ExpIntegralEi[-x]*(1 - ExpIntegralEi[-x]), {x, 0, 200}, WorkingPrecision -> digits + 5]^2 // Join[{0}, RealDigits[#, 10, digits][[1]]]&
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
STATUS
approved