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A272427
Asymptotic variance (normalized by n^2) of the third longest cycle in a random permutation on n symbols.
0
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, 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, 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
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 World of Mathematics, Golomb-Dickman Constant
FORMULA
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.
EXAMPLE
0.00449392318179080474794492205756996926493197843077072420750592398...
MATHEMATICA
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]]]&
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
STATUS
approved