Limit[(n!*LaguerreL[n,-1]/(n^(n+1/4)/Sqrt[2]/E^(n-2*Sqrt[n]+1/2))-1)*Sqrt[n],n->Infinity]

Mathematica wrong output is 13/16 = 0.8125

Right result is 31/48 = 0.645833...

But numerically is computed right (after ~ 1 hour):

N[Table[(n!*LaguerreL[n,-1]/(n^(n+1/4)/Sqrt[2]/E^(n-2*Sqrt[n]+1/2))-1)*Sqrt[n],{n,1000000,10000000,1000000}],20]

{0.64595327485857865704, 0.64591815870538845793, 0.64590259799022250717, 0.64589332088298149047, 0.64588698941420000820, 0.64588231547803760347, 0.64587868275613942973, 0.64587575441366589107, 0.64587332876408306831, 0.64587127669377150813}

I already reported this bug in 2012, but still was not fixed (in versions 7,8,9,10)
http://code.google.com/p/mathematica/issues/list
see Issue 46

For more please see my article 
"Too many errors around coefficient C1 in asymptotic of sequence A002720"
http://members.chello.cz/kotesovec/math_articles/kotesovec_too_many_errors_A002720.pdf