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

0,1

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

Table of n, a(n) for n=0..100.

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

Cf. A143301, A246848.

Sequence in context: A303326 A085967 A163960 * A143531 A352473 A211269

Adjacent sequences: A246846 A246847 A246848 * A246850 A246851 A246852

Keyword

nonn,cons

Author

Jean-François Alcover, Sep 05 2014

Status

approved