A272414 Asymptotic variance (normalized by n^2) of the second longest cycle in a random permutation on n symbols.

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

Table of n, a(n) for n=0..98.

Xavier Gourdon, Combinatoire, Algorithmique et Géométrie des Polynomes Ecole Polytechnique, Paris 1996, page 152 [in French]

Eric Weisstein's MathWorld, Golomb-Dickman Constant

Wikipedia, 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

Cf. A084945, A247398, A272413.

Sequence in context: A115522 A362138 A190283 * A194367 A078312 A004582

Adjacent sequences: A272411 A272412 A272413 * A272415 A272416 A272417

Keyword

nonn,cons

Author

Jean-François Alcover, Apr 29 2016

Status

approved