A272430 Asymptotic variance (normalized by n^2) of the second largest connected component in a random mapping on n symbols.

0, 1, 8, 6, 2, 0, 2, 2, 3, 3, 0, 6, 7, 8, 1, 3, 8, 8, 7, 2, 1, 4, 0, 6, 5, 7, 0, 3, 6, 2, 3, 4, 3, 1, 5, 0, 4, 3, 1, 9, 3, 5, 6, 0, 1, 4, 4, 9, 5, 7, 4, 9, 9, 8, 2, 3, 1, 8, 4, 2, 5, 9, 1, 9, 9, 9, 2, 8, 1, 2, 3, 3, 6, 1, 8, 7, 8, 5, 3, 1, 2, 2, 6, 5, 3, 0, 2, 3, 5, 7, 0, 3, 1, 1, 2, 3, 1, 6, 5

Offset: 0

Author

Jean-François Alcover, Apr 29 2016

Keywords

• nonn
• cons

Examples

			0.01862022330678138872140657036234315043193560144957499823184259199928...
		

References

• Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.4.2 Random mapping statistics, p. 290.

Links

• Xavier Gourdon, Combinatoire, Algorithmique et Géométrie des Polynomes, Ecole Polytechnique, Paris 1996, page 152 (in French).
• Eric Weisstein's World of Mathematics, Flajolet-Odlyzko Constant

Crossrefs

Cf. A084945, A143297, A261873, A272429.

Programs

Mathematica

digits = 98; Ei = ExpIntegralEi; (8/3)*NIntegrate[x*(1 - E^(Ei[-x]/2)*(1 - Ei[-x]/2)), {x, 0, 200}, WorkingPrecision -> digits + 5] - 4*NIntegrate[1 - E^(Ei[-x]/2)*(1 - Ei[-x]/2), {x, 0, 200}, WorkingPrecision -> digits + 5]^2 // Join[{0}, RealDigits[#, 10, digits][[1]]]&

Formula

(8/3)*integral_{0..infinity} x*(1 - e^(Ei(-x)/2)*(1 - Ei(-x)/2)) dx - 4*(integral_{0..infinity} 1 - e^(Ei(-x)/2)*(1 - Ei(-x)/2) dx)^2, where Ei is the exponential integral.