A246879 - OEIS

%I #44 Dec 28 2020 07:30:29

%S 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,

%T 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,

%U 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

%N 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.

%C See A087132.

%H Steven R. Finch, <a href="http://arxiv.org/abs/2001.00578">Errata and Addenda to Mathematical Constants</a>, p. 42.

%H Ph. Flajolet, É. Fusy, X. Gourdon, D. Panario, N. Pouyanne, <a href="http://arxiv.org/abs/math/0606370">A Hybrid of Darboux's Method and Singularity Analysis in Combinatorial Asymptotics</a>, arXiv:math/0606370 [math.CO]

%F prod_{k>=1} I_0(2/k), where I_0 is the zeroth modified Bessel function.

%e 4.2634035141526697782989350551661966905350818174794...

%p evalf(product(BesselI(0,2/k), k=1..infinity), 100) # _Vaclav Kotesovec_, Sep 17 2014

%t 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

%Y Cf. A087132.

%K nonn,cons

%O 1,1

%A _Jean-François Alcover_, Sep 08 2014

%E More terms from _Vaclav Kotesovec_, Sep 17 2014