A246849 Decimal expansion of 1-delta_0, where delta_0 is the Hall-Montgomery constant (A143301).

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

Offset: 0

Author

Jean-François Alcover, Sep 05 2014

Keywords

• nonn
• cons

Comments

This constant, by coincidence, is also a limiting probability concerning the number of cycles of a given length in a random permutation.

One has P_1(xi) = 1-delta_0 = Pi^2/6 - log(xi) - log(xi)^2 - 2*Li_2(xi), where xi = 1/(1+sqrt(e)) (see A246848 and the references).

Examples

			0.82849950685846393413956002844478789037773709577...
		

Links

• Steven R. Finch, Errata and Addenda to Mathematical Constants, arXiv:2001.00578 [math.HO], 2020, p. 29.
• Michael Lugo, The number of cycles of specified normalized length in permutations, arXiv:0909.2909 [math.CO], 2009.
• Eric Weisstein's MathWorld, Hall-Montgomery Constant

Crossrefs

Cf. A143301, A246848.

Programs

Mathematica

Pi^2/6 + Log[1 + Sqrt[E]] - Log[1 + Sqrt[E]]^2 - 2*PolyLog[2, 1/(1 + Sqrt[E])] // RealDigits[#, 10, 101]& // First

PARI

Pi^2/6 + log(exp(1/2)+1) - log(exp(1/2)+1)^2 - 2*polylog(2, 1/(exp(1/2)+1)) \\ Charles R Greathouse IV, Sep 08 2014

Python

from mpmath import mp, log, exp, polylog, pi
mp.dps=102
print([int(n) for n in list(str(pi**2/6 + log(exp(1/2)+1) - log(exp(1/2)+1)**2 - 2*polylog(2, 1/(exp(1/2)+1)))[2:-1])]) # Indranil Ghosh, Jul 04 2017

Formula

Pi^2/6 + log(1 + sqrt(e)) - log(1 + sqrt(e))^2 - 2*Li_2(1/(1 + sqrt(e))), where Li_2 is the dilogarithm function.