A246879 Decimal expansion of the constant W(1) appearing in the asymptotic expression of the probability that two independent, random n-permutations have the same cycle type as W(1)/n^2.
4, 2, 6, 3, 4, 0, 3, 5, 1, 4, 1, 5, 2, 6, 6, 9, 7, 7, 8, 2, 9, 8, 9, 3, 5, 0, 5, 5, 1, 6, 6, 1, 9, 6, 6, 9, 0, 5, 3, 5, 0, 8, 1, 8, 1, 7, 4, 7, 9, 4, 1, 1, 6, 0, 5, 0, 6, 7, 7, 1, 2, 5, 6, 3, 2, 0, 3, 7, 1, 9, 1, 4, 5, 8, 2, 7, 8, 5, 7, 3, 4, 6, 1, 7, 2, 3, 5, 6, 1, 3, 4, 4, 8, 1, 3, 2, 9, 8, 7, 7, 3, 0, 6, 3, 5
Offset: 1
Examples
4.2634035141526697782989350551661966905350818174794...
Links
- Steven R. Finch, Errata and Addenda to Mathematical Constants, p. 42.
- Ph. Flajolet, É. Fusy, X. Gourdon, D. Panario, N. Pouyanne, A Hybrid of Darboux's Method and Singularity Analysis in Combinatorial Asymptotics, arXiv:math/0606370 [math.CO]
Crossrefs
Cf. A087132.
Programs
-
Maple
evalf(product(BesselI(0,2/k), k=1..infinity), 100) # Vaclav Kotesovec, Sep 17 2014
-
Mathematica
digits = 50; m0 = 1000; dm = 1000; tail[m_] := PolyGamma[1, m] - (1/24)*PolyGamma[3, m] + PolyGamma[5, m]/1080 - (11*PolyGamma[7, m])/967680 + (19*PolyGamma[9, m])/217728000 - (43*PolyGamma[11, m])/94058496000; Clear[f]; f[m_] := f[m] = Sum[Log[BesselI[0, 2/k]], {k, 1, m - 1}] + tail[m] // N[#, digits + 5] &; f[m0]; f[m = m0 + dm]; While[RealDigits[f[m], 10, digits + 2] != RealDigits[f[m - dm], 10, digits + 2], Print["f(", m, ") = ", f[m]]; m = m + dm];RealDigits[Exp[f[m]], 10, digits] // First
Formula
prod_{k>=1} I_0(2/k), where I_0 is the zeroth modified Bessel function.
Extensions
More terms from Vaclav Kotesovec, Sep 17 2014
Comments