OFFSET
0,1
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).
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
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.
EXAMPLE
0.82849950685846393413956002844478789037773709577...
MATHEMATICA
Pi^2/6 + Log[1 + Sqrt[E]] - Log[1 + Sqrt[E]]^2 - 2*PolyLog[2, 1/(1 + Sqrt[E])] // RealDigits[#, 10, 101]& // First
PROG
(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
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Jean-François Alcover, Sep 05 2014
STATUS
approved