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

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

Table of n, a(n) for n=0..99.

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

Wikipedia, 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

Cf. A084945, A247398, A272413, A272414, A272415.

Sequence in context: A021961 A115365 A263491 * A068340 A097667 A202388

Adjacent sequences: A272424 A272425 A272426 * A272428 A272429 A272430

Keyword

nonn,cons

Author

Jean-François Alcover, Apr 29 2016

Status

approved